'^1*w 


"!lOV> 


Digitized  by  tiie  Internet  Arciiive 

in  2007  witii  funding  from 

IVIicrosoft  Corporation 


littp://www.arcliive.org/details/elementsofnavigaOOIiendiala 


ELEMENTS    OF    NAVIGATION 


[See  p.  59 


NAVIGATOR  TAKING  A   SIGHT 


Elements  of  Navigation 

A    complete    Exposition    of    the 

Newest  Methods  as  used  in  the 

Navy  and  Merchant  Marine 


BY 
W.    J.    HENDERSON,   A.M. 

FORMERLY  LIEUTENANT  N.M.N.Y.  AND    NOW 

INSTRUCTOR     IN     NAVIGATION 

NEW  YORK  NAVAL  MILITIA 


AN     ENTIRELY     NEW     EDITION 

REWRITTEN  THROUGHOUT  AND 

BROUGHT    UP  TO    DATE 


HARPER  &  BROTHERS  PUBLISHERS 

NEW   YORK   AND    LONDON 


Books  by 
W.  J.  HENDERSON 


ELEMENTS   OF   NAVIGATION 

Illustrated.     12mo,  Cloth 
AFLOAT    WITH    THE    FLAG 

Illustrated.     Post  8vo,  Cloth,  Ornamental 
SEA  YARNS   FOR   BOYS,  Spun  by  an  Old  Salt 

Illustrated.     Post  8vo,  Cloth,  Ornamental 


HARPER  &   BROTHERS,  NEW  YORK 


Elements  of  Mavigation 


Copyright,   1895,   1918.   by  Harper  &  Brothers 

Printed  in  the  United   States  of  America 

Published  July,  1918 


VK 

55  5 
H3g 


TO 

Commodore    RobERT     P.    FoRSHEW 

COMMANDING     THE     NAVAL     MILITIA 
N  EW     YORK 


CONTENTS 


FAQE 

Preface ix 

The  Compass 1 

Variation 9 

Deviation 12 

Azimuths  of  the  Sun 13 

Use  of  the  Pelorus 18 

Leeway 19 

The  Log 21 

The  Lead-line 25 

Charts 27 

Chart  Sailing  or  Piloting 33 

Dead-reckoning 38 

Working  a  Traverse 48 

Hove  To 52 

The  Day's  Run 53 

Shaping  the  Course 55 

Navigation  by  Observation 57 

The  Sextant 58 

Sextant  Adjustments 62 

Index  Error       64 

Hints  on  Taking  Altitudes 65 

Correcting  the  Altitude 65 

The  Chronometer 69 

The  Nautical  Almanac 71 

Apparent  and  Mean  Time — the  Equation  .  74 


CONTENTS 

PAGE 

Latitude  by  Meridian  Altitude     ....  76 

Use  of  Latitude  Constant 80 

Meridian  Altitude  of  a  Star 82 

Meridian  Altitude  of  a  Planet    ....  84 

Meridian  Altitude  op  the  Moon  ....  85 

Meridian  Altitude  Below  the  Pole       .     .  87 

Conversions  of  Arc  and  Time 89 

Latitude  by  Ex-meridian  Altitude  ....  92 

Sidereal  Time  and  Right  Ascension       .     .  94 

Latitude  by  the  Pole-star 98 

Time  Azimuths 99 

Longitude  by  Time  Sights  op  Sun      .     .     .  101 

Longitude  Sights  of  Stars 104 

Sumner's  Method 108 

The  St.  Hilaire  Method 124 

St.  Hilaire  Meridian  Formula       ....  129 

Great-circle  Sailing 134 

Distance  and  Danger  Angles 138 

Allowance  for  Tides 142 

Rating  a  Chronometer 144 

Care  op  a  Chronometer 146 

The  Day's  Work 149 

Compensation  of  the  Compass 154 

Finding  the  Deviation 159 

The  Napier  Curve 102 

Examples  for  Practice 166 


PREFACE 

This  is  literally  a  new  edition,  inasmuch 
as  the  book  has  been  rewritten.  The  science 
of  navigation  has  beer,  materially  modified 
in  recent  years  and  the  practice  of  the  navy 
has  introduced  changes  looking  toward  ce- 
lerity in  arriving  at  results.  Formula  which 
a  few  years  ago  were  regarded  as  of  prime 
importance  have  been  relegated  to  secondary 
places,  and  some  entirely  abandoned,  while 
new  ones  have  assumed  the  leading  positions. 

The  author  has  therefore  remodeled  this 
entire  work.  All  that  was  of  permanent 
worth  in  the  edition  j^receding  this  has  been 
retained,  but  some  of  it  has  been  amplified 
with  the  purpose  of  making  the  explanations 
still  clearer. 

Special  stress  has  been  laid  on  the  necessity 
of  learning  all  the  systems  of  marking  com- 
pass bearings  and  courses  and  making  the 
corrections  for  variation  and  deviation.  Azi- 
muths of  the  sun  are  introduced  at  this 
point,  but  other  methods  of  finding  deviation 
are  postponed  till  later  in  the  book. 


PREFACE 

The  treatment  of  dead  reckoning  has  been 
expanded  and  the  working  of  the  traverse 
illustrated  in  the  full  navy  form  as  now  used. 
The  recommencement  of  dead  reckoning  from 
observations  in  the  course  of  the  day's  work 
is  explained. 

The  sections  dealing  with  latitude  sights 
have  been  rewritten  and  arranged  in  a  dif- 
ferent order  and  the  use  of  the  prepared  con- 
stant is  thoroughly  explained  and  illustrated. 

While  ex-meridians  are  retained  the  "Phi 
Prime  and  Phi  Second"  sight  is  omitted. 
It  has  no  longer  any  practical  value  since 
the  St.  Hilaire  method  accomplishes  all  that 
it  used  to.  The  treatment  of  all  sidereal 
work  in  this  edition  is  newly  written,  and  the 
former  explanation  of  sidereal  time  and  right 
ascension  made  clearer.  Relations  of  hour 
angles  and  right  ascensions  are  illustrated 
fully  by  diagrams. 

The  use  of  the  haversine  in  time  sights 
supersedes  that  of  the  sine  of  apparent  time. 

The  Sumner  method  is,  of  course,  accorded 
its  proper  importance  and  new  diagrams  in- 
troduced, but  the  section  dealing  with  the 
St.  Hilaire  method  has  been  greatly  en- 
larged. In  it  the  author  has  introduced  a 
formula  of  his  own  for  applying  the  St. 
Hilaire  principle  to  meridian  sights.  This 
formula  has  been  submitted  to  the  Hydro- 
graphic  Office  and  pronounced  correct. 
Graphic  diagrams  are  presented. 


PREFACE 

The  section  on  the  navigator's  routine  at 
sea  has  been  rewritten  entirely  with  a  view 
to  bringing  it  more  into  conformity  with  the 
present  naval  practice. 

A  chapter  on  the  compensation  of  the 
compass  has  been  added,  with  detailed  in- 
structions as  to  how  to  find  and  record 
deviations,  how  to  adjust  all  corrections,  and 
how  to  make  and  use  the  Napier  curve. 

Finally,  everything  in  the  book  has  been 
arranged  so  as  to  instruct  the  student  in  the 
navy  way  of  doing  things  on  the  ships  now 
at  sea.  The  work  is  in  complete  harmony 
with  Bowditch,  but  its  explanations  are  de- 
signed to  meet  the  needs  of  students  who 
are  not  astronomers  nor  mathematicians. 

The  author  is  gratified  at  the  reception 
accorded  to  the  revised  editions  published 
in  1917.  But  since  the  United  States  entered 
the  war  he  has  been  engaged  in  instructing 
naval  militia  and  naval  reserve  officers  and 
has  profited  by  the  experience.  It  is  his 
confident  belief  that  the  present  edition  will 
be  found  a  marked  improvement  on  its  pred- 
ecessors. 

The  author's  thanks  are  due  to  Rear- 
Admiral  Seton  Schroeder,  Hydrographer, 
Navy  Department,  for  his  courteous  con- 
sideration of  the  work  mentioned  under  St. 
Hilaire  Method;  and  he  also  owes  a  debt  of 
gratitude  to  his  capable  colleague,  Mr.  Frank 
Seymour  Hastings,  for  examination  and  cor- 


PREFACE 

rection  of  all  illustrative  problems  which 
have  been  worked  up  for  the  present  year. 
Thanks  are  also  due  to  Mr,  John  Bliss,  of 
John  Bliss  &  Co.,  the  well-known  compass 
adjuster,  for  assistance  in  preparing  the  mat- 
ter on  compass  compensation. 


ELEMENTS    OF   NAVIGATION 


ELEMENTS  OF   NAVIGATION 


THE   COMPASS 

^  Navigation  is  the  art  of  finding  the  geo- 
graphical location  of  a  vessel  at  sea,  the  most 
direct  course  to  be  steered  in  pursuit  of  the 
voyage,  and  the  distance  to  be  made. 

There  are  two  branches  of  the  art — dead- 
reckoning  and  observation. 

Navigation  by  dead-reckoning  consists  in 
actually  measuring  the  courses  and  distances 
sailed  by  the  ship,  and  from  them  computing 
the  distance  and  direction  from  the  port  left 
and  to  the  port  sought, 

Navigation  by  observation  consists  in  com- 
puting the  position  of  the  ship  by  the  appli- 
cation of  astronomical  and  mathematical  laws. 

The  problems  of  dead-reckoning  are  solved 
by  plane  trigonometry;  those  of  observation 
by  spherical  trigonometry.  But  as  the  trigo- 
nometrical data  are  all  provided  in  the  tables 
printed  in  epitomes  of  navigation,  the  mari- 
ner is  not  required  to  be  acquainted  with 


2       ELEMENTS  OF  NAVIGATION 

any  higher  methematics  than  simple  arith- 
metic. 

The  instruments  used  in  dead-reckoning  are 
'^  \  the  compass,  log,  and  lead-line.  The  com- 
'  pass  shows  the  direction  in  which  the  ship 
is  traveling;  the  log  measures  the  speed  or  the 
distance.  The  lead  is  used  when  on  soundings 
to  measure  the  depth  of  water  and  ascertain 
the  character  of  the  bottom.  These  data, 
referred  to  in  the  chart,  throw  valuable  light 
on  the  question  of  the  ship's  position.  Ap- 
proaching a  coast  in  thick  weather,  or  on  a 
,i  dark,  cloudy  night,  the  lead  is  the  navigator's 
'[    main  reliance. 

In  addition  to  these  instruments,  the  navi- 
gator requires  for  all  his  work  accurate  charts 
of  the  waters  which  he  is  traversing  and  their 
coasts.  Charts  issued  by  governments  are 
more  trustworthy  than  those  published  by 
private  firms,  which  have  not  the  resources 
of  nations. 

The  mariner's  compass  is  the  first  instru- 
ment which  the  navigator  must  know.  It  is 
presumed  that  any  person  who  reads  this 
book  has  seen  a  compass;  therefore  it  is  not 
described.  The  card  is  the  part  which  con- 
cerns the  learner  at  this  point.  Its  circum- 
ference is  divided  into  32  equal  parts,  called 
points.  Each  point  has  a  name;  all  these 
names  the  student  must  learn. 

Any  intelligent  person  can  easily  discover 
the  system  on  which  the  points  are  named. 


THE  COMPASS  3 

North,  south,  east,  and  west  are  called  the 
cardinal  points;  northeast,  southeast,  south- 
west, and  northwest  are  the  intercardinal 
points.    Each  cardinal  point  is  8  points  away 


COMPASS-CABD,    SHOWLNQ    POINTS    AND    DEQBEES 


from  the  nearest  cardinal,  and  4  points 
away  from  the  nearest  intercardinal.  The 
student  must  learn  to  repeat  the  names  in 
order  from  north  to  east  and  around  to  north 
again,  thus: 


4      ELEMENTS   OF   NAVIGATION 

North,  north  -  by  -  east,  north  -  northeast, 
northeast-by-north,  northeast,  northeast-by- 
east,  east-northeast,  east-by-north,  east,  east- 
by-south,  east-southeast,  southeast-by-east, 
southeast,  southeast-by-south,  south-south- 
east, south-by-east,  south,  south-by-west, 
south-southwest,  southwest-by-south,  south- 
west, southwest  -  by  -  west,  west  -  southwest, 
west-by-south,  west,  west-by-north,  west- 
northwest,  northwest  -  by  -  west,  northwest, 
northwest-by-north,  north-northwest,  north- 
by-west,  north. 

This  is  called  "boxing  the  compass." 
In  reckoning  courses  by  their  names  we  al- 
ways count  from  the  north  and  south  line 
of  the  compass,  which  is  called  the  meridian. 
Thus  north  -  northeast,  south  -  southeast, 
north  -  northwest,  and  south  -  southwest  are 
2-point  courses.  East  and  west  are  8-point 
courses.  Southeast-by-south  is  a  3-point 
course.  The  student  should  learn  how 
many  courses  of  each  kind  there  are,  bear- 
ing in  mind  that  there  is  nothing  greater 
than  an  8-point  course.  After  a  careful 
study  of  the  points  the  student  should  be 
able  to  answer  with  facility  all  such  ques- 
tions as  these : 

How  many  1-point  courses  are  there? 
2-point?  3-point,  etc.?  Name  them.  How 
many  points  is  it  from  E.S.E.  to  S.W.-by- 
S.?  How  many  points  from  N.E.-by-E.  to 
W.-by-S.?     How  many  points  from  E.-by- 


THE  COMPASS  5 

S.  to  E.N.E.?  What  points  are  3  points 
away  from  W.-by-N.? 

Until  the  student  is  master  of  the  points 
of  the  compass  and  their  relations  he  should 
go  no  further.  When  he  has  learned  them, 
he  must  acquaint  himself  with  the  half  and 
quarter  points  as  set  forth  in  the  following 
tables.    (See  pages  6  and  7.) 

The  next  step  is  to  learn  the  angle  which 
each  course  makes  with  the  meridian.  Merid- 
ians are  imaginary  lines  running  from  the 
north  to  the  south  pole  and  used  for  deter- 
mining longitude.  The  meridian  of  the  com- 
pass is  its  north-and-south  line,  which  for 
the  moment  may  be  assumed  to  coincide 
with  a  longitude  meridian.  A  ship  sailing 
northeast  makes  a  course  at  an  angle  of  45° 
from  the  meridian,  and  the  compass  shows 
us  tliis. 

The  circumferences  of  all  circles,  no  matter 
what  their  size,  are  divided  into  360  equal 
parts  called  degrees.  All  angles  are  measured 
by  these.  These  facts  are  of  hourly  use  in 
navigation. 

For  example,  in  the  diagram  (see  page 
8)  the  angles  at  a  do  not  increase  in 
size  because  their  boundary  lines  are  pro- 
longed. They  will  measure  45°  on  any 
circle,  even  the  earth's  circumference.  A 
degree,  therefore,  is  -sh  of  any  circle's  cir- 
cumference. 

If  you  divide  the  360°  of  the  compass-card 


6      ELEMENTS   OF   NAVIGATION 

by  its  32  points,  you  will  learn  that  1  point 
equals  11°  15'.     By  adding  11°  15'  for  each 


TABLE    SHOWING    THE    NAMES    OP     POINTS     AND     QUARTER-POINTS, 

AND  THE  ANGLE  MADE  BT 


North 

Points 

N.ME. 

N.MW. 

Ya. 

2° 

48'  45" 

N.HE. 

N.i^W. 

K 

5° 

37'  30" 

N.ME. 

N.J'iW. 

Ya. 

8° 

26'  15" 

N.-bv-E. 

N.-by-W. 

1 

11° 

15'  — 

N.-by-E.ME. 

N.-by-W.MW. 

IM 

14° 

3'  45" 

N.-by-E.i^E. 

N.-by-W.i-^W. 

\Y2 

10° 

52'  30" 

N.-by-E.34E. 

N.-by-W.  J4W. 

\% 

19° 

41'   15" 

N.N.E. 

N.N.W. 

2 

22° 

30'  — 

N.N.E.^E. 

N.N.W.MW. 

2K 

2,5° 

18'  45" 

N.N.E. i^E. 

N.N.W.HW. 

2H 

28° 

7'  30" 

N.N.E. ME. 

N.N.W.^W. 

254 

30° 

56'  15" 

N.E.-by-N. 

N.W.-by-N. 

3 

33° 

45'  — 

N.E.34N. 

N.W.JiN. 

3}^ 

30° 

33'  45" 

N.E.J^N. 

N.W.HN. 

3J-'2 

39° 

22'  30" 

N.E.MN. 

N.W.i^N. 

3M 

42° 

11'  15" 

N.E. 

N.W. 

4 

45° 

—    — 

N.E.ME. 

N.W.14W. 

4}i 

47° 

48'  45" 

N.E.J^E. 

N.W.i^W. 

^Yi 

50° 

37'  30" 

N.E.ME. 

N.W.MW. 

4  5  4 

53° 

26'  15'' 

N.E.-by-E. 

N.W.-bv-W. 

5 

56° 

15'  — 

N.E.-by-E.i4E. 

N.W.-by-W.MW. 

5M 

59° 

3'  45" 

N.E.-by-E.i^E. 

N.W.-by-W.i^W. 

oH 

61° 

52'  30" 

N.E.-by-E.^E. 

N.W.-by-W.?-iW. 

5fi 

64" 

41'  15" 

E.N.E. 

W.N.W. 

6 

67° 

30'  — 

E.N.E.^E. 

W.N.W.MW. 

%% 

70° 

18'  45" 

E.N.E.  l^E. 

W.N.W.MW. 

6H 

73° 

7'  30" 

E.N.E.J^E. 

W.N.W.s^W. 

6?i 

75° 

56'  15" 

E.-by-N. 

W.-by-N. 

7 

78° 

45'  — 

E.?iN. 

W.3iN, 

7M 

81° 

33'  45" 

E.HN. 

W.J^N. 

71^ 

84° 

22'  30" 

E.KN. 

W.^N. 

734 

87° 

11'  15" 

East. 

West. 

8 

90° 

—    — 

additional  point  you  will  learn  that  2  points 
equal  22°  30';  3  points,  33°  45';  4  points,  45°; 
5  points,  56°  15';  6  points,  67°  30';  7  points, 


THE  COMPASS  7 

78°  45';  and  8  points,  90°.     Sailing-vessels 
cannot  be  steered  closer  than  a  quarter  of  a 


NUMBER  OF  POINTS  AND  FRACTIONS  OF  POINTS  IN   EACH  COURSE, 
EACH  WITH  THE  MERIDIAN. 


South 

Points 

S.ME. 

S.MW. 

M 

2°  48'  45" 

S.HE. 

S.^W. 

H 

5°  37'  30" 

S.»4E. 

S.JiW. 

H 

8°  26'  15" 

S.-by-E. 
S.-by-E. ME. 

S.-by-W. 

1 

ll"  15'  — 

S.-by-W.^W. 

IH 

14°     3'  45" 

S.-by-E.  I'zE. 

S.-by-W.  i^W. 

IH 

10°  52'  30" 

S.-by-E.  J4E. 

S.-by-W.'>4W. 

IM 

19°  41'   15" 

S.S.E. 

S.S.W. 

2 

22°  30'  — 

S.S.E.ME. 

S.S.VV.MW. 

2M 

25°  18'  45" 

S.S.E.KE. 

S.S.W.i^W. 

2,^ 

28°     7'  30" 

S.S.E.ME. 

8.S.W.HW. 

2M 

30°  50'   15" 

S.E.-by-S. 

S.W.-by-S. 

3 

33°  45'  — 

S.E.HS. 

S.W.34S. 

3M 

30°  33'  45" 

S.E.3^8. 

S.W.J^^S. 

'SH 

39°  22'  30" 

S.E.^S. 

S.W.MS. 

3M 

42°  11'   15" 

S.E. 

s.w. 

4 

45°  —     — 

S.E.ME. 

S.w.MW. 

4M 

47°  48'  45" 

S.E.  HE. 

s.w.i^w. 

4V2 

50°  37'  30" 

S.E.ME. 

S.W.MW. 

4M 

53°  26'  15" 

S.E.-by-E. 

S.W.-by-W. 

5 

50°  15'  — 

S.E.-by-E.^E. 

S.W.-by-W.MW. 

r,i4 

59°     3'  45" 

S.E.-by-E.  J^E. 

S.W.-by-W.i^W. 

^M 

61°  52'  30" 

S.E.-by-E.ME. 

S.W.-by-W.MW. 

5M 

04°  41'   15" 

E.S.E. 

W.S.W. 

6 

07°  30'  — 

E.S.E.ME. 

W.S.W.MW. 

6M 

70°  18'  45" 

E.S.E.J^E. 

W.S.W.J^W. 

6H 

73°    7'  30" 

E.S.E.HE. 

W.S.W.MW. 

6M 

75°  56'  15" 

E.-by-S. 

W.-by-S. 

7 

78°  45'  — 

E.JiS. 

W.MS. 

7M 

81°  33'  45" 

E.HS. 

W.J^S. 

7^ 

84°  22'  30" 

E.HS. 

W.MS. 

7M 

87°  11'  15" 

East. 

West. 

8 

90° 

point,  and  for  their  navigation  a  quarter  may 
be  called  3°.  Steamers  can  be  steered  to  de- 
grees, and  their  courses  are  so  set.     They 


8       ELEMENTS   OF   NAVIGATION 

may  be  expressed  as  so  many  degrees  east 

or  west  from  the  meridian,  thus:  N.  47°  E., 

or  S.  36°  W. 

—--^«^.  The    latest    method, 

^~^v,  however,  and   that  al- 

X  ways  used  in  the  navy, 

/   \  is  to  count  the  degrees 

/^\       \,^      all  the  way  around  from 

/       \^      \°      N.  by  way  of  E.  and  S. 

/  y       \      back    to     N.,    and    set 

'h. 1        1      courses     accordingly. 

The    count    runs    from 

MEASUREMENT  OF  ANGLES  ^O        ^^        ggQO       ^^^        ^^^_ 

graphical  direction  is 
omitted,  so  that  courses  are  expressed  simply 
as  being  of  so  many  degrees,  as,  for  example, 
46°,  137°,  220°. 

Every  navigator  should  master  thoroughly 
the  relations  of  courses  reckoned  in  one  way 
to  the  same  reckoned  in  another.  For 
example,  he  should  know  that  every  course 
up  to  89°  would  be  found  in  the  northeast 
quadrant  of  the  compass;  those  between  90° 
(E.)  and  180°  (S.)  in  the  southeast;  those 
between  180°  (S.)  and  270°  (W.)  in  the  south- 
west, and  those  between  270°  and  360°  in  the 
northwest. 

How  express  S.  27°  W.  in  360°  system? 
S.  =  180°.    This  +  27°=  207°. 

How  express  S.  27°  E.?    180°-  27°=  153°. 

How  express  N.  27°  W.?    360°  -  27°  =  333°. 

How  express  W.-by-N.?    270°+ 11°  =  281°. 


VARIATION  9 

How  express  intercardinal  points?   N.  E.  = 
45°;S.E.  =  135°;  S.W.=  225°;  N.W.=  315°. 


VARIATION 

The  north  point  of  the  compass  indi- 
cates true  or  geographical  north  at  only  a 
few  places  on  the  globe.  At  all  other  places 
it  points  a  little  to  one  side  or  the  other  of 
north.  This  error  is  called  variation  of  the 
compass. 

It  is  caused  by  the  fact  that  the  magnetic 
north  and  south  poles  of  the  earth  do  not 
coincide  with  the  true  or  geographical  poles. 
The  former  is  several  hundred  miles  south 
of  the  geographical  pole,  and  the  latter  sev- 
eral hundred  miles  north.  The  needle  is 
perfectly  true;  it  points  right  at  the  magnetic 
north  pole.  But  that  pole  is  not  the  north 
end  of  the  earth's  axis. 

In  navigating  a  vessel  it  is  necessary  to 
make  allowance  for  this  variation.  The 
amount  of  allowance  and  its  direction  are 
indicated  on  the  charts.  On  large  charts, 
such  as  that  of  the  North  Atlantic,  will  be 
found  irregular  lines  marked  10°  W,,  15° 
W.,  etc.  This  means  that  along  this  line 
the  variation  of  the  compass  from  true  north 
is  10°  W.,  15°  W.  There  are  certain  lines 
which  have  no  variation,  and  here  no  allow- 
ance is  to  be  made.     On  small  charts,  such 


10     ELEMENTS   OF   NAVIGATION 

as  that  of  New  York  Bay,  the  variation  is 
shown  by  the  compass-card  printed  on  the 
chart.  The  north  point  of  it  will  be  found 
slewed  a  little  to  the  eastward  or  westward 
of  a  meridian  line,  and  near  it  will  be  seen 
an  inscription,  such  as  "Variation  11°  W. 
in  1892."  Now  let  us  see  how  this  variation 
affects  the  compass  aboard  ship,  and  how 
we  are  to  allow  for  it,  so  that  we  shall  know 
exactly  which  way  we  are  going. 

Let  the  outer  circle  represent  the  sea  hori- 
zon, the  inner  circle  the  compass-card.  The 
variation  is  one  point  westerly.  Hence  the 
north  point  of  the  compass  points  to  the 
north-by-west  point  of  the  horizon,  and  the 
south  point  of  the  compass  to  the  south-by- 
east  point  of  the  horizon.  In  other  words, 
standing  at  the  center  and  looking  toward 
the  circumference,  you  find  that  every  point 
on  the  compass  is  one  point  to  the  left  of  the 
proper  place.  If  your  compass  says  you  are 
sailing  north,  you  are  really  sailing  N.-by-W. 
If  its  says  south,  you  are  going  S.-by-E. 
If  it  says  east,  you  are  going  E.-by-N.  Hence 
we  get  these  rules : 

To  correct  a  compass  course. — ^When  the 
variation  is  westerly,  the  true  course  will 
be  as  many  points  to  the  left  of  the  compass 
course  as  there  are  points  of  variation* 
When  the  variation  is  easterly,  the  true 
course  will  be  as  many  points  to  the  right 
of  the  compass  course. 


VARIATION 


11 


Conversely,  having  ascertained  the  true 
course  between  two  places,  you  must  con- 
struct the  required  compass  course  by  apply- 
ing the  variation  in  a  direction  the  reverse 


TrueMorffi 


VARIATION    OK    COMPASS 


of  that  used  in  converting  a  compass  to  a 
true  course. 

To  convert  inic  course  to  compass  course. — 
Variation  westerly,  compass  course  to  right 
of  true  course.  Variation  easterly,  compass 
course  to  left  of  true  course. 

When  working  courses  reckoned  from  1° 
to  360°,  westerly  variation  is  called  — ,  and 


12     ELEMENTS   OF   NAVIGATION 

easterly  -f.  To  convert  compass  to  true 
course,  subtract  amount  of  W.  var.;  add 
amount  of  E.  var.  To  convert  true  to  com- 
pass course,  reverse  the  process,  adding  W. 
var.,  and  subtracting  E. 

A  course  or  bearing  affected  only  by  var. 
(and  not  by  deviation)  is  called  magnetic. 


DEVIATION 

In  addition  to  the  force  of  terrestrial  mag- 
netism, which  affects  all  compasses  alike,  no 
matter  how  situated,  we  have  to  contend 
with  deviation,  which  is  error  caused  by  the 
influence  of  neighboring  iron  or  steel.  In 
ships  built  of  either  inetal  the  influence  is 
great  and  no  compass  aboard  such  a  ship 
is  ever  quite  correct,  except  possibly  on  one 
or  two  courses.  When  the  ship  changes  her 
course,  the  hull  assumes  a  new  relation  to  the 
direction  of  the  needles  of  the  compass,  and 
hence  the  deviation  changes. 

Therefore  it  becomes  necessary  to  know 
the  error  on  each  course.  Compasses  are 
compensated  by  the  use  of  magnets,  which 
reduce  error  to  the  miniinum,  but  some  always 
remains.  A  fuller  explanation  of  the  cause, 
the  nature,  and  the  treatment  of  deviation 
will  be  given  in  a  chapter  on  compass  com- 
pensation. At  present  it  is  only  necessary 
to  note  that  when   the  compass   has  been 


AZIMUTHS  OF  THE  SUN         13 

compensated,  a  table  of  the  residual  errors  is 
made  for  the  information  of  the  navigator. 
Since,  however,  the  deviations  are  Hable  to 
change  in  voyages  involving  much  alteration 
of  latitude,  the  tables  cannot  be  too  im- 
plicitly trusted. 

Deviation  is  ascertained  on  every  fifteenth 
degree  of  the  circumference  of  the  compass, 
and  the  table  of  residual  errors  would  begin 
thus: 


Ship's 

Head  by  Standard  Compa.s.s 

Deviation 

North .... 

0° 
15.° 
30° 
45» 
60° 
75° 
90° 

-  14°  27' 

-  12°  la' 

-  9°  38' 

-  7°   12' 

-  6°  02' 

-  4°  55' 

-  3°  10' 

N.  E 

E 

Deviations  are  named  easterly  (+)  or 
westerly  (  — ),  just  as  variation  is,  and  the 
correction  is  applied  by  the  same  rules. 


AZIMUTHS    OF   THE    SUN 

Deviations  are  found  at  sea  by  what  are 
called  azimuths  of  celestial  bodies.  The 
process  consists  in  taking  a  compass  bearing 
of  the  object  and  comparing  it  with  the  true 
bearing.  The  true  bearing  at  any  time  may 
be  computed  from  the  altitude  and  declina- 
tion of  the  body,  and  the  latitude  of  the  ob- 


14     ELEMENTS   OF   NAVIGATION 

server.  To  save  labor  we  have  sets  of  azimuth 
tables  giving  the  true  bearing  of  the  sun  for 
every  ten  minutes  of  the  day. 

The  most  familiar  instrument  for  taking 
the  compass  bearing  is  the  azimuth  mirror, 
a  circular  contrivance  which  fits  over  the 
compass  and  has  sight  vanes  for  observing 
the  body  and  a  mirror  which  throws  a  re- 
flected beam  of  light  on  the  compass-card, 
thus  showing  the  bearing. 

Having  taken  the  compass  bearing  and 
ascertained  the  local  time  of  the  observation, 
enter  the  azimuth  tables  (Hydrographic 
Office,  Book,  No.  71),  with  the  declination 
above  and  the  time  at  the  side.  You  must 
seek  the  page  marked  at  the  top  with  your 
latitude,  and  note  that  the  book  is  divided 
into  two  parts,  lat.  and  dec.  of  same  name, 
and  lat.  and  dec.  of  different  names.  Be 
careful  to  select  the  proper  page  and  then 
pick  out  the  true  bearing  to  the  nearest  de- 
gree. Azimuths  are  read  from  N.  toward  E. 
or  W.  when  you  are  in  N.  lat.,  and  S.  to  E. 
or  W.  in  S.  lat.;  thus,  N.  120°  W.,  S  .60°  E. 
After  picking  them  out  in  this  form,  you  must 
convert  the  reading  to  that  of  a  compass  bear- 
ing.    N.  120°  W.=  S.  60°  W.,  or  240°. 

The  difference  between  the  compass  and 
the  true  bearings  is  the  total  error  of  the 
compass.  The  difference  between  the  total 
error  and  the  variation  as  given  by  the  chart 
is  the  deviation. 


AZIMUTHS  OF  THE  SUN 


15 


How  to  determine  the  correct  local  time, 
and  how  to  use  other  celestial  bodies,  will 
be  explained  further  on.  The  best  way  of 
computing  the  deviation  is  to  find  the  diff. 
between  comp.  and  true  bearings,  and  then 
between  error  and  variation. 


Compass  bearing . 
True  bearing 

N.  75°  E. 
N.  fjo"  E. 

Error 

Var 

20°  W. 
10°  W. 

Dev 

10°  w. 

Compass  bearing.  ..  S.  15°  W. 

True  beiirinK S.  20°  W. 

Error 5°  E. 

Var 10°  W. 

Dev 15°  E. 


When  the  courses  are  set  in  degrees  in  the 
360°  system,  the  formula  for  correcting  a 
compass  course  is: 

T.  C.=  C.  C.+ Var.+ Dev. 

T.  C.  means  true  course;  C.  C,  compass 
course.  Easterly  var.  or  dev.  is  plus;  west- 
erly is  minus.  If  both  quantities  have  the 
same  sign,  add  the  two  and  prefix  the  sign. 
If  the  quantities  have  different  signs,  sub- 
tract the  less  from  the  greater  and  prefix  the 
sign  of  the  greater.  .  Apply  resultant  quan- 
tities to  C.  C.  in  each  case  and  obtain  T.  C. 

Examples: 

1. — Compass  course,  195°.  Variation,  20° 
W.  Deviation,  5°  W.  Required,  true  course. 
Westerly  error  is  a  minus  quantity,  hence; 


Var.        -20° 
Dev.       -  5" 

Error      -25«* 


C.  C.        195» 
Error    -  25° 


T.  C. 


170° 


16     ELEMENTS   OF   NAVIGATION 

2 .— C.  C,  195°.  Var.,  20°  E.  Dev.,  5°  E. 
Required,  T.  C.  Easterly  error  is  always  a 
plus  quantity.    Hence : 


Var. 
Dev. 


+  20° 
+  5° 


Error      +25° 


C.  C.        195° 
Error      +25° 


T.  C. 


3.— C.  C,  195°.    Var.,  20°  E.    Dev.,  5°  W. 
Required  T.  C. 


Var. 
Dev. 


+  20° 
—   5° 


Error      +15° 


C.  C.        195° 
Error      + 15° 


T.  C. 


210° 


4.— C.  C,  195°.    Var.,  20°  W.    Dev.,  5°  E. 
Required,  T.  C. 


Var. 
Dev. 


-20° 
+   5° 


Error      — 15° 


C.  C.        195° 
Error    -    15° 


T.  C. 


180° 


Given  the  T.  C.,  Var.,  and  Dev.  to  find 
the  C.  C.  to  be  steered,  the  navigator  re- 
verses the  former  process  by  changing  the 
signs  prefixed  to  easterly  and  westerly  errors. 
Easterly  becomes  minus  and  westerly  plus. 

Examples: 

1.— T.  C.  170°.  Var.,  20°  W.  Dev.,  5°  W. 
Required,  C.  C. 


Var. 
Dev. 


Cor. 


+20° 
+  5° 


+  25° 


T.  C.        170° 
Cor.      +   25° 


c.  c. 


195° 


AZIMUTHS  OF  THE  SUN         17 

2  — T.  C,  180°.    Var.,  20°  W.    Dev.,  5°  E. 
Required,  C.  C. 

Var.        +20°  T.  C.        180° 

Dev.       -   5°  Cor.      +    15° 


■      Cor.        +15°  C.  C.        195° 

The  student  will  note  that  these  are  reverse 
workings  of  the  first  and  fourth  examples  of 
^e  correction  of  a  C.  C.  to  find  T.  C. 
and  they  bring  us  back  to  our  former  C.  C. 
A  little  practice  will  convince  the  student 
that  the  new  method  is  easier  than  the  old. 

Keep  all  loose  iron  and  steel  as  far  as  pos- 
sible from  your  compasses.  Bear  in  mind 
that  magnetic  influence  will  not  be  stopped 
])y  placing  anything  between  the  compass  and 
the  iron  or  steel.  It  will  pass  through  a  stone 
wall. 

Make  it  an  invariable  rule  to  ascertain  the 
deviation  of  the  compass  on  every  course 
steered  and  to  correct  the  course  accordingly. 

Bear  in  mind  when  ascertaining  your  devi- 
ation that  it  is  good  only  for  that  one  course. 
If  your  ship  is  heading  E.S.E.  and  you  find 
the  deviation  to  be  10°  E.,  it  will  be  some- 
thing else  the  moment  you  alter  the  course 
to  E.-by-S.,  or  even  E.S.E.3^E. 

Bear  in  mind  in  taking  bearings  to  ap- 
ply the  deviation  according  to  the  direction  of 
the  ship^s  head. 

For  instance,  you  are  lying  at  anchor. 
Your   compasses   have  just   been   adjusted. 


18     ELEMENTS   OF   NAVIGATION 

The  ship's  head  points  N.W.-by-N.  The 
table  of  errors  says  that  on  that  course  the 
deviation  is  one  point  easterly.  Directly  on 
your  starboard  beam  is  a  light  house.  You 
wish  to  get  its  bearing.  The  compass  says 
it  bears  N.E.-by-E.  But  you  have  one  point 
easterly  deviation.  Hence  the  correct  com- 
pass bearing  is  E.N.E. 

Large  vessels  carry  more  than  one  com- 
pass. One  of  these  is  situated  above  the 
deck  and  as  far  away  from  local  influences 
as  possible.  It  is  called  the  standard  com- 
pass, and  the  ship  is  navigated  by  it. 

To  set  a  course  by  a  standard  compass. — 
Stand  by  the  standard  yourself  and  station 
a  man  at  the  steering  compass.  Order  the 
helm  to  port  or  starboard  till  the  ship  is 
precisely  on  her  course  by  the  stancfard. 
At  that  instant  blow  a  whistle  (or  give  any 
other  preconcerted  signal),  and  the  man  at 
the  steering  compass  notes  the  direction  of 
the  ship's  head  according  to  it.  The  course 
which  he  gets  is  the  one  to  be  given  to  the 
helmsman. 

USE    OF   THE    PELORUS 

In  ascertaining  deviations  and  in  all  other 
operations  requiring  the  taking  of  bearings 
the  pelorus  will  be  found  useful.  The  in- 
strument is  a  type  of  dumb  compass,  which 
may  be  set  up, in  any  convenient  place.    It 


LEEWAY  19 

has  an  outer  ring  of  brass  with  the  degrees 
marked  thereon,  and  within  this  is  a  dumb- 
compass  card  of  ground  glass.  Outside  of 
all  revolves  a  pair  of  sight  vanes  through 
which  bearings  are  taken. 

If  now  the  fore-and-aft  line  of  the  dumb 
card  is  made  to  coincide  with  the  course  of 
the  ship,  bearings  taken  by  pelorus  will  be  the 
same  as  those  taken  by  the  compass.  If 
the  deviation  is  known  and  is  eliminated,  the 
pelorus  may  be  set  accordingly,  and  all  bear- 
ings will  be  magnetic.  If  the  variation  and 
deviation  are  both  eliminated,  bearings  by 
pelorus  will  be  true. 


LEEWAY 

Leeway  is,  of  course,  not  an  error  of  the 
compass;  but  as  it  has  to  be  considered  in 
the  correction  of  compass  courses  in  dead- 
reckoning,  it  is  convenient  to  introduce  the 
subject  here.  A  sailing-vessel  on  a  wind,  or 
even  with  the  wind  abeam,  will  slide  off  to 
leeward  more  or  less.  A  strong  wind  will 
affect  even  a  steamer.  Consequently  her 
actual  course  will  not  be  that  indicated  by 
compass,  even  when  corrected  for  variation 
and  deviation. 

To  find  the  leeway. — Experienced  sailors 
can  estimate  the  leeway  by  the  angle  between 
the  vessel's  wake  and  her  keel.    A  good  plan, 


20     ELEMENTS   OF   NAVIGATION 

however,  is  to  heave  the  log,  then  bring  the 
line  to  the  center  of  the  compass,  and  its 
angle  with  the  vessel's  course  will  show  the 
amount  of  leeway. 

To  correct  for  leeway. — Leeway  on  the  star- 
board tack  is  the  same  as  westerly  variation. 
Leeway  on  the  port  tack  is  the  same  as 
easterly  variation.  The  corrections  are  made 
in  the  same  way.     A  glance  at  the  diagram 


DIAGRAM    OF    LEEWAY 


will  make  this  clear.  The  vessel  heading 
N.E.  on  the  starboard  tack  and  making  a 
quarter-point  of  leeway  is  actually  going 
N.E.J^N.  The  vessel  on  the  port  tack  head- 
ing N.W.  and  making  a  quarter-point  of  lee- 
way is  really  going  N.W.i<4N. 

A  good  point  to  remember  is  this:  lee- 
way on  the  port  tack  and  westerly  varia- 
tion or  deviation  are  opposed  to  one  another, 
and  the  same  is  true  of  leeway  on  the  star- 
board tack  and  easterly  error.    For  example, 


THE  LOG 


21 


you  have  a  quarter-point  westerly  variation, 
no  deviation,  and  a  quarter-point  leeway  on 
the  port  tack;  the  leeway  and  variation 
counterbalance  one  another,  and  the  com- 
pass course  is  the  true  course.  The  form 
given  in  the  following  examples  for  practice 
is  used  in  computing  dead-reckoning: 


Compass 
Course 

Vari- 
ation 

Deviation 

Leeway 

True  Course 

S.W.-by-W. 

E.-by-S. 
N.N.E.HE. 

S.  42°  E. 

S.  33°  W. 
227° 

J^pt.W. 
16°  W. 
1  pt.  E. 
20°  W. 
r,°  E. 
10°  W. 

Mpt.W. 

10°  E. 
2  pts.  W. 

2.5°  E. 

3°  W. 

4°  W. 

Mpt.Port 
J^pt.Btar. 
Mpt.Star. 
6°  Port 
3°  Star. 
4°  Star. 

S.W.J^W. 

E.i^S. 

N.?iE. 
S.  31°  E. 
S.  32°  W. 

209° 

THE   LOG 


There  are  two  kinds  of  logs,  the  chip 
log  .and  the  patent  or  taffrail  log.  The 
principal  parts  of  the  chip  log  are  the  chip, 
the  reel,  the  line,  and  the  toggle.  A  second- 
glass  is  used  for  measuring  the  time.  The 
chip  is  a  triangular  piece  of  wood,  rounded 
on  its  lower  edge  ^nd  ballasted  with  lead 
to  make  it  ride  point  up.  The  toggle  is  a 
little  wooden  case  into  which  a  peg,  joining 
the  ends  of  the  two  lower  lines  of  the  bridle, 
is  set  in  such  a  way  that  a  jerk  on  the  line 
will  free  it,  causing  the  log  to  lie  flat  so  that 
it  can  be  hauled  in.    The  inboard  end  of  the 


22     ELEMENTS   OF   NAVIGATION 

line  is  wound  around  the  reel.  The  first 
10  or  15  fathoms  of  line  from  the  log-chip 
are  called  "stray  line,"  and  the  end  of  this 
is  distinguished  by  a  mark  of  red  bunting 
6  inches  long.  Its  purpose  is  to  let  the  chip 
get  clear  of  the  swirl  under  a  vessel's  counter 
before  reckoning  begins. 


Line 


CHIP    LOG    AND    REEL 


The  knots,  as  they  are  called,  are  dis- 
tinguished by  running  pieces  of  fish-line 
through  the  strands  to  the  number  of  one, 
two,  three,  etc.  A  piece  of  white  bunting, 
two  inches  long,  marks  every  two-tenths  of 
a  knot.  This  is  because  the  run  of  a  ship  is 
recorded  in  knots  and  tenths. 

A  new  log-line  should  be  soaked  in  water 


THE  LOG  23 

a  few  days  before  marking,  and  always  before 
leaving  port  you  should  soak  your  line  and 
then  see  that  the  marks  are  all  at  the  proper 
distances. 

The  log-glass,  in  appearance  like  an  hour- 
glass, measures  28  seconds.  For  high  rates 
of  speed,  a  14-second  glass  is  used,  and  then 
the  number  of  knots  shown  by  the  line  must 
be  doubled.  In  damp  weather  a  watch  is 
better  than  a  sand-glass. 

The  principle  of  the  chip  log  is  that  the 
length  of  a  knot  bears  the  same  ratio  to 
the  nautical  mile  (6,080  feet)  as  the  time  of 
the  glass  does  to  the  hour.  /Hence  we  get 
this  proportion :  / 

3600*:  6080  :  :  28  sec.  :x 
X  =  47  feet  9  inches. 

The  speed  of  the  ship  is  recorded  in  the  log- 
book in  knots  and  tenths  of  a  knot. 

How  to  heave  the  chip  log. — Have  an  assist- 
ant to  hold  the  glass.  See  that  all  the  sand 
is  in  the  bottom.  Heave  the  log-chip  well 
out  to  leeward  from  the  stem,  and  hold  the 
reel  so  the  line  will  run  freely.  As  soon  as 
the  stray  line  is  out  call  "Turn,"  and  the 
assistant  must  turn  the  glass  quickly  and 
start  the  sand  running.  The  instant  the 
sand  has  passed   down  the  assistant  must 

*  Number  of  seconds  in  an  hour. 


24     ELEMENTS   OF   NAVIGATION 

call  "Stop,"  and  you  check  the  line.  Note 
the  number  of  knots  and  tenths  and 
haul  in. 

The  chip  log  should  be  hove  every  hour. 
If  the  speed  varies  between  hours  it  must 
be  estimated,  or  the  log  hove  again. 

The  patent  or  towing  log  consists  of  a 
dial,  a  line,  and  a  rotator  of  screw-propel- 


PATENT    OR    TOWING    LOG 


ler  form.  The  action  of  the  water  on  the 
rotator,  which  is  at  the  end  of  the  line  and 
thrown  overboard,  causes  the  line  to  make 
a  certain  number  of  twists  a  minute.  These 
twists  are  proportional  to  the  speed  of  the 
vessel,  and  they  move  the  machinery  of  the 
dial,  which  records  miles  and  fractions  of  a 
mile. 


THE   LEAD  LINE  25 

In  setting  a  taffrail  log  to  work,  you  must 
note  where  the  dial  stands  at  the  time  when 
you  throw  over  the  rotator.  The  reading  of 
the  log  is  noted  in  the  log-book  once  an  hour, 
and  whenever  the  course  is  changed.  It 
should  also  be  read  when  an  observation  is 
taken. 

Both  logs  are  liable  to  error.  The  rotator 
of  the  patent  log  slips  sometimes,  and  that 
underrates  the  distance  gone.  Usually,  how- 
ever, it  overrates.  The  chip  log  is  likely  to 
underrate  with  a  following  sea,  which  causes 
the  chip  to  "come  home,"  and  to  overrate 
a  little  with  a  head  sea. 

In  shallow  water,  but  out  of  sight  of  land- 
marks, a  vessel  drifting  in  a  tideway  may 
use  a  ground  log.  This  is  a  common  log- 
line  \vith  a  hand  lead  attached,  and  it  shows 
the  actual  speed  of  the  ship  over  the  ground. 


THE   LEAD-LINE 

The  lead  is  used  to  ascertain  the  depth 
of  water,  and,  when  necessary,  the  character 
of  the  bottom.  There  are  two  kinds  of 
leads:  the  hand  lead  and  deep-sea  lead.  The 
first  weighs  from  7  to  14  lbs.,  and  has  mark- 
ings to  20  fathoms.  The  second  weighs  from 
80  to  150  lbs.,  and  is  used  in  depths  over 
100  fathoms.  The  hand  lead  is  marked 
thus: 


26     ELEMENTS   OF   NAVIGATION 


2  fathoms,  2  strips  of  leather. 

3  "  3       ■• 

5  "  a  whit '  rag. 

7  "  a  red  r^is;. 

10  "  a  piece  of  leather  with  a  hole  in  it. 

13  "  sam.?  33  at  3. 

15  "  "         "     5. 

17  "  "         "     7. 

20  "  with  2  knots. 


Large  hand  leads  and  deep-sea  leads  are 
marked  above  20  fathoms  with  an  addi- 
tional knot  at  every  10-fathom  point  (30, 
40,  50,  etc.),  and  a  single  knot  at  each  inter- 
vening 5-fathom  point  (25,  35,  45,  etc.). 

Deep-sea  leads  are  hollowed  out  on  the 
lower  end  so  that  an  "arming"  of  tallow  can 
be  put  in.  This  will  bring  up  a  specimen  of 
the  bottom,  which  should  be  compared  with 
the  description  found  on  the  chart. 

All  first-class  sea-going  vessels  should  dis- 
card the  deep-sea  lead  for  Lord  Kelvin's 
sounding-machine.  This  apparatus  consists 
of  a  cylinder  around  which  are  wound  about 
300  fathoms  of  piano  wire.  To  the  end  of 
this  is  attached  a  heavy  lead.  An  index 
on  the  side  of  the  instrument  records  the 
number  of  fathoms  of  wire  paid  out.  Above 
the  lead  is  a  copper  cylindrical  case  in  which 
is  placed  a  glass  tube  open  only  at  the  bot- 
tom and  ground  inside.  The  pressure  of  the 
sea  forces  water  up  into  this  tube,  as  it  goes 
down,  a  distance  proportionate  to  the  depth, 
and  the  ground  part,  being  wet,  shows  clear. 
When  hoisted,  the  tube  is  laid  upon  a  pre- 


CHARTS  27 


pared  scale,   and  the  height  to  which   the 
water  has  been  forced  inside  shows  the  depth/ 
in  fathoms  on  this  scale. 


CHARTS 

A  chart  is  a  map  of  an  ocean,  bay,  sound, 
or  other  navigable  water,  showing  the  con- 
formation of  the  coasts,  heights  of  moun- 
tains, the  depth  at  low-water,  direction  and 
velocity  of  tidal  currents,  location,  charac- 
ter, height  and  radius  of  visibility  of  all 
beacon  Ughts,  location  of  rocks,  shoals,  and 
buoys,  and  nature  of  the  bottom  wherever 
soundings  can  be  obtained. 

The  top  of  the  chart  is  generally  north. 
If  for  any  reason  it  is  otherwise,  north  will 
be  indicated  by  the  north  point  of  a  com- 
pass-card printed  somewhere  on  the  chart. 

On  the  majority  of  small  charts,  such  as 
those  of  bays,  harbors,  and  sounds,  the  com- 
pass on  the  chart  includes  the  variation; 
that  is,  its  north  point  is  slewed  east  or  west, 
just  as  that  of  a  real  compass  (without  devi- 
ation) would  be  in  that  place.  In  laying  off 
courses  by  such  a  compass  you  do  not  have 
to  allow  for  variation,  because  it  is  already 
allowed  for.  On  large  charts,  such  as  that  of 
the  North  Atlantic,  the  compass  is  printed 
true,  and  the  variation  is  indicated  by  lines 
as  described  under  the  head  of  "Variation." 


28     ELEMENTS   OF   NAVIGATION 

Parallels  of  latitude  are  shown  by  straight 
lines  across  the  chart.  The  degrees  and 
minutes  are  marked  on  the  perpendicular 
border. 

Meridians  of  longitude  are  shown  by- 
straight  lines  up  and  down  the  chart,  and 
the  degrees  and  minutes  are  recorded  on  the 
horizontal  border. 

The  navigator  should  know  the  varieties 
of  buoys.  Channels  on  the  United  States 
coasts  are  indicated  by  red  buoys  with  even 
numbers  situated  on  the  starboanl  side  com- 
ing in  from  the  sea,  and  black  buoys  with 
odd  numbers  on  the  port  side. 

Buoys  with  black-and-white  perpendicu- 
lar stripes  are  in  mid-channel  and  must  be 
passed  close  to. 

Buoys  with  red-and-black  horizontal  stripes 
indicate  obstructions  with  channels  on  both 
sides. 

The  abbreviations  on  charts  are  easily 
understood. 

Soundings  on  plain  white  usually  are  in 
fathoms,  especially  in  general  charts,  and 
those  on  shaded  parts,  in  feet.  In  charts  of 
small  bodies,  such  as  New  York  Bay,  sound- 
ings are  often  all  in  feet. 

To  avoid  error  in  this  and  other  matters 
read  carefully  all  text  printed  on  the  chart. 
It  is  there  for  your  information. 

On  general  charts  of  coasts  there  are  fathom 
curves,  showing  the  lines  along  which  run 


CHARTS  29 

soundings  of  10,  20,  30,  etc.,  fathoms.  These 
give  valuable  aid  to  the  coastwise  navigator, 
as  well  as  to  him  approaching  the  coast  from 
the  ocean. 

A  light  is  indicated  by  a  red  and  yellow 
spot.  F.  means  fixed;  Fl.,  flashing;  Int., 
intermittent;  Rev.,  revolving,  etc. 

An  arrow  indicates  a  current  and  its  direc- 
tion.   The  speed  is  always  recorded. 

Rocks  just  under  water  are  shown  by  a  cross 
surrounded  by  a  dotted  circle;  rocks  above 
water,  by  a  dotted  circle  with  dots  inside  it. 

The  charts  used  by  mariners,  except  in 
great-circle  sailing,  are  called  Mercator's 
charts.  Speaking  roughly,  this  chart  is  con- 
structed on  the  imaginary  theory  that  the 
earth  is  cylindrical.  Hence  the  meridians 
of  longitude,  which  in  a  sphere  (see  page  30) 
converge  at  the  poles,  are  opened  out  and 
become  straight,  parallel  lines.  This  compels 
a  stretching  out  in  width  of  everything  repre- 
sented in  high  latitudes.  To  preserve  the 
geographical  relations  the  length  is  also 
stretched  proportionately,  so  that  although 
everything  in  high  latitudes  is  on  too  large 
a  scale  as  compared  with  places  in  lower 
latitudes,  the  courses  and  distances  measured 
on  a  chart  are  correct.  The  advantage  of  a 
chart  made  in  this  way  is  that  it  enables  the 
course  of  a  ship  to  be  represented  by  a 
straight  line,  whereas  on  a  sphere  it  would  be 
— and  truthfully  so — a  curved  one. 


30     ELEMENTS   OF   NAVIGATION 


/ 

/ 

/ 

— If 

SPHERE    COMPABbU    WITH    MERCATOR's    CHART 


CHARTS  31 

In  very  high  latitudes  the  inexactness  of 
a  Mercator's  chart  reveals  itself  fully.  It 
is  quite  impracticable  for  polar  navigation. 
For  instance,  how  can  you  steer  for  the 
north  pole  on  a  chart  whose  meridians  never 
come  together  at  any  pole,  but  are  infinitely 
prolonged  parallel  lines?  Owing  also  to  this 
inexactness  the  bearings  of  distant  objects 
are  not  always  quite  correct  when  laid  down 
in  straight  lines  on  the  chart.  But,  taking  it 
all  in  all,  the  Mercator's  chart  is  the  one  best 
adapted  to  the  daily  needs  of  the  mariner. 

By  means  of  the  chart  the  navigator  may 
at  times  sail  along  a  coast  in  clear  weather 
without  having  recourse  to  any  other  in- 
struments of  navigation  than  the  compass 
and  lead-line. 

The  instruments  used  in  consulting  the 
chart  are  the  pacallel  rules,  dividers,  and 
course-protractor. 

The  parallel  rules  are  made  of  ebony  or 
gutta-percha.  They  are  connected  by  cross- 
pieces  of  brass,  working  on  pivots  in  su€h 
a  way  that  the  rules  may  be  spread  apart 
or  pushed  together,  but  will  always  remain 
parallel  to  each  other. 

They  are  used  to  determine  the  direction 
of  courses.  For  instance,  you  wish  to  find 
the  course  from  Sandy  Hook  Lightship  to 
Fire  Island  Light.  Lay  the  parallel  rules 
so  that  one  edge  cuts  both  places.  Now  slide 
first  one  rule  and  then  the  other,  holding  the 
3 


32     ELEMENTS   OF   NAVIGATION 


unmoved  one  down  firmly  so  as  to  retain 
the  direction,  till  the  edge  cuts  the  center  and 
circumference  of  the  compass  printed  on  the 
chart.  The  edge,  if  the  direction  has  been 
preserved,  will  indicate  the  course. 

The  dividers  are  used  to  measure  distance. 
On  small  charts  take  your  dis- 
tance from  the  scale  of  nauti- 
cal miles;  on  large  ones,  from 
the  latitude  scale  at  the  side  of 
the  chart.  A  minute  of  lati- 
tude is  always  a  mile,  because 
parallels  of  latitude  are  equi- 
distant at  all  parts.  A  minute 
of  longitude  is  a  mile  only  at 
the  equator,  for  the  meridians 
are  always  coming  nearer  and 
nearer  together,  till  at  the  pole 
they  join  and  there  is  no  lon- 
gitude at  all.  Yet,  as  every 
parallel  of  latitude  runs  all  the  way  around 
the  earth,  it  is  a  circle  and  contains  360°.  The 
distance  from  A  to  B  will  be  the  same  num- 
ber of  degrees,  minutes,  and  seconds  whether 
measured  on  parallel  A  or  parallel  E,  but  it 
will  not  be  the  same  number  of  miles.  But 
the  distances  from  A  to  C,  from  C  to  D, 
and  from  D  to  E  must  be  the  same  on  any 
meridian,  because  the  lines  A,  C,  D,  and  E 
are  parallel.  That  is  why  distance  is  meas- 
ured on  the  latitude  scale. 

Long  courses  are  most  conveniently  shaped 


PARALLEL    RULES 


CHART  SAILING  OR  PILOTING   33 

by  the  course-protractor,  Tliis  consists  of  a 
long  single  rule  upon  which  slides  a  movable 
disk  marked  as  a  compass-card.     By  laying 


^.^*^7 

r^^ 

,/V// 1 

\\\^ 

,////  /  1 

clll  1    1 

\    \   \  \\ 

III  1  I 

MINUTE3    VERSUS    MILES 


the  rule  down  on  the  course  and  bringing 
the  north  point  of  its  disk  to  coincide  with  a 
meridian,  the  angle  of  the  course  is  at  once 
seen.  Variation  can  be  allowed  for  in  placing 
the  disk's  north  point,  if  so  desired. 


CHART    SAILING    OR    PILOTING 


These  titles  cover  various  methods  of 
locating  the  ship  and  ascertaining  distances 
sailed. 

Finding  position  by  cross-hearings. — Select 
two  charted  objects  whose  bearings  from  the 
ship  will  be  at  right  angles  to  each  other, 
or  nearly  so.  Take  an  accurate  bearing  of 
each.      Correct    bearings    for    the    deviation 


34     ELEMENTS   OF   NAVIGATION 


Ship  ^ 


known  to  exist  on  the  heading  of  the  ship,  not 
on  the  direction  of  the  bearings.  With  the 
parallel  rules  (applied  to  compass-card  on 
chart)  lay  off  the  two  lines  of  bearing  with 
light  pencil-marks  on  chart.  Where  they  in- 
tersect will  be  the  ship's  position. 

The  ship's  position  may  be  accurately  de- 
termined by  measuring 
with  a  sextant  the  horizon- 
tal angles  (see  Distance 
and  Danger  Angles)  sep- 
arating three  charted  of)- 
jects.  and  plotting  the  po- 
;;ition  with  a  three-armed 
i>rotractor.  This  is  a 
metal  ring  marked  with 
degrees  and  having  three 
arms,  one  fixed,  two  mov- 
able. The  three  arms  can 
be  set  to  the  angles 
measured  by  sextant  and 
then  laid  down  on  the 
charted  objects.  The  cen- 
ter of  the  instrument  will  be  at  the  ship's 
position. 

To  find  the  distance  between  two  places  on 
the  chart. — If  the  course  is  due  north  or  south, 
measure  the  distance  and  refer  it  to  the  lati- 
tude scale  on  the  side  of  the  chart  precisely 
opposite  the  course.  The  number  of  minutes 
in  the  distance  as  found  in  the  scale  will  be 
the  number  of  miles,  because  1'  of  lat.  is  one 


MAP   OF  CROJS-BEARING8 


CHART  SAILING  OR  PILOTING   35 

mile.  If  the  course  is  east  or  west,  follow 
same  rule. 

If  the  course  runs  diagonally,  measure  the 
distance  on  the  latitude  scale  opposite  the 
middle  of  the  course.  The  best  way  is  to 
take  off  the  lat.  scale  with  the  dividers  a  con- 
venient unit,  such  as  two  or  five  miles,  and 
find  how  many  times  it  is  contained  in  the 
distance  between  the  places. 

On  charts  of  small  areas,  such  as  bays, 
use  the  scale  of  nautical  miles  found  on  the 
chart. 

To  find  the  latitude  of  a  place  on  the  chart. — 
Put  one  leg  of  the  dividers  in  the  place  and  the 
other  in  the  nearest  parallel  of  lat.  Apply 
the  dividers  thus  opened  to  the  lat.  scale 
at  side  of  chart,  one  leg  touching  the  same 
parallel  as  before.  The  other  will  be  at  the 
required  lat.  To  find  a  longitude,  do  the  same 
thing,  but  use  a  meridian  and  the  long,  scale 
at  top  or  bottom  of  chart. 

To  mark  the  ship's  place  on  the  chart. — 
This  is  to  be  done  at  sea  after  finding  the 
latitude  and  longitude.  "With  the  di\'iders 
take  from  the  graduated  meridian  the  given 
latitude;  mark  this  on  the  meridian  nearest 
the  given  longitude;  lay  the  edge  of  a  pair 
of  parallel  rulers  on  a  near  parallel,  and  work 
one  side  of  them  to  the  exact  latitude  you 
have  marked  on  the  meridian;  then  with  the 
dividers  take  the  given  longitude  from  the 
graduated  parallel  [at  the  top  or  bottom  of 


36     ELEMENTS   OF   NAVIGATION 


the  chart];  lay  this  down  along  the  edge  of 
the  parallel  rulers  which  already  mark  the 
latitude,  and  you  have  the  ship's  place" 
(Qualtrough). 

To  find  the  ship's  position  when  sailing  along 
the  land. — Take  a  compass  bearing  of  a  hght 
or  other  prominent  object  when  it  is  2,  3,  or 
4  points  off  the  course.  Take  another  bear- 
ing when  it  has  doubled  the  first  and  is  4,  6, 
or  8  points  off  the  course.  The  distance  run 
by  the  ship  between  the  two  bearings  will  be 
her  distance  from  the  observed  object  at  the 
second  bearing. 

In  the  diagram  the  ship  at  A  heading  north 
finds  the  light  bearing 
N.N.W.,  2  points  off  her 
course.  At  B  she  finds 
it  bears  N.W.,  4  points 
off.  The  log  makes  the 
distance  from  A  to  B  7 
miles.  The  distance  of 
the  light  from  the  ship 
at  B  will  be  the  same. 
The  commonest  form  of 
this  problem  is  that  used 
at  positions  B  and  C, 
with  the  object  4  points 
off  the  course  and  ex- 
actly abeam.  This  is 
known  as  the  bow-and- 
beam  bearing.  The  nav- 
coASTwisB  BEARINGS      igator  wlU  fiud  cascs  in 


CHART  SAILING  OR  PILOTING   37 


which  the  other  form  is  convenient.  This 
method  should  be  practised  continually,  as 
it  is  the  standard  method  in  coastwise  navi- 
gation. It  is  also  valuable  in  establishing  a 
final  position  with  reference  to  the  land  when 
about  to  go  to  sea. 

How  to  use  compass,  log,  and  lead  in  a  fog. 
— Take  a  piece  of  tracing-paper  and  rule  a 
meridian  on  it.  Take  casts  of  the  lead  at 
regular  intervals,  noting  the  time  at  which 
each  cast  is  taken,  and  the  distance  logged 
between  each  two.  The 
compass  shows  the 
course.  Now  rule  a  line 
on  the  tracing-paper  in 
the  direction  of  your 
course.  Measure  off  on 
it  by  the  scale  of  miles 
of  your  chart  the  dis- 
tances run  between 
casts.  Opposite  each 
cast  note  the  time  and 
the  depth  ascertained. 
It  is  a  good  thing  to 
add  also  the  character 
of  the  bottom.  Now 
lay  your  tracing-paper 
down  on  the  chart, 
which  can  be  seen  through  it,  in  the  neighbor- 
hood of  the  position  you  believed  yourself  to 
be  in  when  you  made  the  first  cast.  If  your 
chain  of  soundings  agrees  with  those  on  the 


CHAIN    OF  SOUNDINGS 


38     ELEMENTS   OF   NAVIGATION 

chart  right  under  your  course,  all  is  right.  If 
not,  move  the  tracing-paper  about,  keeping  the 
meridian  line  due  north  and  south,  till  you 
find  the  place  on  the  chart  that  does  agree 
with  you.  That  is  where  you  are.  You  will 
not  find  two  places  where  you  can  get  that 
chain  of  soundings  on  the  same  course  and 
at  the  same  distances. 

This  is  the  only  method  by  which  a  ship's 
position  can  be  found  with  any  certainty  on 
soundings  in  thick  weather.  There  is  no 
excuse  whatever  for  the  man  who  runs  his 
vessel  ashore,  if  he  has  not  tried  this. 


DEAD-RECKONING 

To  ascertain  the  position  of  a  ship  at  sea 
by  keeping  account  of  the  courses  and  dis- 
tances which  she  sails,  or  by  "dead-reckon- 
ing," we  proceed  on  the  theory  that  small 
sections  of  the  surface  of  the  earth  are  flat. 
The  whole  matter  then  resolves  itself  into 
the  solution  of  right-angled  triangles.  A 
single  glance  will  show  the  student  that  any 
of  the  courses  ruled  on  the  diagram  chart 
unite  with  the  parallels  and  meridians  in 
forming  series  of  right-angled  triangles.  The 
only  cases  in  which  no  such  triangles  exist 
are  those  of  sailing  due  east  and  west  or  due 
north  and  south. 

The  problems  to  be  solved  in  sailing  on 


DEAD-RECKONING 


39 


the  open  sea  out  of  sight  of  land  are  these: 
Having  left  a  known  point  and  sailed  so 
many  miles  in  such  and  such  direction,  what 
latitude  and  longitude  have  we  arrived  at, 


80° 


70° 


40° 


^^ 

==j    ,^ 

,,--'' 

^^"' 

k 

\  '' 

s. 

W"^ 

•>            "'^r 

>^ 

\ 

^, 

■'■ 

\ 

"\, 

\^ 

40° 


30° 


20° 


10° 


DIAGRAM    CHART 


and  what  are  the  course  and  distance  thence 
to  our  point  of  destination? 

If  you  are  sailing  due  north  or  south,  the 
problem  is  extremely  simple.  Suppose  your 
position  at  noon  to-day  is  lat.  41°  15'  N., 
long.  40°  W.,  and  up  to  noon  to-morrow  you 
sail  280  miles  north  (true).  It  is  obvious 
that  the  longitude  will  remain  unchanged. 


40     ELEMENTS   OF   NAVIGATION 

The  latitude  will  be  280  minutes,  or  4°  40', 
farther  north.  That  4°  40'  is  called  the  dif- 
ference of  latitude,  and  in  this  case  it  is  ob- 
viously to  be  added  to  to-day's  latitude, 
because  we  have  been  increasing  our  latitude. 
The  ship's  position  at  to-morrow  noon,  then, 
is  lat.  45°  55'  N.,  long.  40°  W. 

The  distance  by  which  a  ship  changes  her 
latitude  north  or  south  is  called  difference  of 
latitude. 

Let  us,  then,  formulate  the  rule. 

To  find  the  new  latitude. — If  the  old  lati- 
tude (called  latitude  left)  and  the  diff.  lat. 
are  of  the  same  name  (both  N.  or  both  S.) 
it  is  obvious  that  j^ou  are  increasing  your 
latitude.  Therefore  add  and  name  sum  N. 
or  S.  after  the  old  latitmile.  If  lat.  left  and 
diff.  lat.  are  of  different  names  you  are  de- 
creasing your  latitude.  Hence  subtract  and 
name  accordingly.  In  either  case  you  ob- 
tain the  new  latitude,  sometimes  called 
lat.  in.  \ 

In  sailing  due  east  or  west,  however,  the 
matter  is  not  so  simple,  because  only  on 
the  equator  are  a  nautical  mile  and  a  minute 
of  longitude  the  same  thing.  But  if  we  have 
a  table  giving  us  the  number  of  miles  in  a 
degree  of  longitude  at  every  distance  north 
or  south  of  the  equator  (which  means  in 
every  latitude),  we  can  easily  find  the  longi- 
tude. For  instance,  a  ship  in  lat.  42°  N.  sails 
true  east  100  miles;  how  much  does  she  alter 


DEAD-RECKONING  41 

her  longitude?  A  degree  of  longitude  in  lat. 
40°  measures  44.59  miles.  She  changes  her 
longitude  by  2°  10.8'  or  2°  10'  48"— a  tenth 
of  a  minute  being  G". 

The  number  of  miles,  then,  which  a  ship 
makes  east  or  west  is  called  departure,  and 
it  must  be  converted  into  degrees,  minutes, 
and  seconds  in  order  to  find  the  difference 
of  longitude. 

But  nine  times  out  of  ten  a  ship  sails  a 
diagonal  course.  Suppose  a  vessel  in  lat. 
40°  20'  N.,  long.  60°  15'  W.,  sails  53  miles 
S.W.-by-W.3^W.  How  are  we  to  find  her  new 
latitude  and  longitude?  She  has  sailed  a 
course  like  this:.  Suppose  we  draw  a  per- 
pendicular line  to  rep- 
resent a  meridian,  and 
a  horizontal  one  to  rep- 
resent a  parallel.  Then 
we  shall  have  the  tri- 
angle ABC,  in  which 
the  line  AC  represents 
the  distance  and  direction,  while  the  angle  at 
A  is  the  angle  of  the  course  with  the  meridian. 
If  now  we  can  ascertain 
the  length  of  AB,  or  the 
distance  l)y  which  she  has 
gone  to  the  south,  we  shall 
have  the  difference  of  lati- 
tude; and  if  we  can  get 
the  length  of  the  line  BC,  we  shall  have  the 
departure  and  from  it  the  difference  of  longi- 


42     ELEMENTS   OF   NAVIGATION 

tude.     From  these  two  factors  we  get  the 
new  latitude  and  longitude. 

This  is  a  simple  problem  in  trigonometry, 
but  no  navigator  need  know  trigonometry, 
because  Tables  1  and  2  of  Bowditch  solve 
all  possible  problems  of  this  kind  for  him, 
and  he  needs  only  arithmetic. 

The  complete  Navigation  Tables  can  be 
purchased  separate  from  the  rest  of  the 
work,  under  the  title  Useful  Tables,  for  $2.25. 

Table  1  is  marked  at  the  top  with  the 
different  courses  from  }4  point  up  to  4  points. 
In  three  adjoining  columns  are  found  distance, 
difference  of  latitude,  and  departure,  marked 
Dist.,  Lat.,  and  Dep.  If  you  are  sailing  on 
any  particular  course,  say  N.N.E.,  you  go 
to  the  table  for  2-point  courses, 
look  in  the  distance  column  for  o^^sn 
the  distance  you  have  made  by  \ 
your  log,  and  opposite  to  that  \ 
distance  you  will  find  your  diff.  \\3; 
lat.  and  dep.  ^\ 

At  4  points  diff.  of  lat.  and  dep.  \ 
become  equal,  becduse  the  course  \ 
is  precisely  half-way  be- 
tween  no  points  and  8     ^^ Dv>*smiM, 

points.     On  any  course 

less  than  4  points  diff. 

lat.  is  greater  than  dep., 

because    you    go    more 

north  or  south  than  east  or  west.     On  any 

course  greater  than  4  points  dep.  is  greater  than 


.-0-^ 


DEAD-RECKONING  43 

diff .  lat.,  because  you  go  more  east  or  west  than 
north  or  south.  And  the  relations  of  the  two 
elements  are  simply  reversed,  as  may  be  seen 
by  the  diagrams.  In  a  2-point  course,  the 
diff.  lat.  is  the  same  as  the  dep.  in  a  6-point 
course,  the  complement  of  a  2-point  course. 
Hence,  in  using  the  tables,  as  soon  as  you 
have  a  course  over  4  points,  you  begin  at  the 
last  page  of  the  tables  and  read  up  from  the 
bottom,  noting  that  while  dist.  remains  in  the 
same  place,  lat.  and  dep.  arc  reversed. 

Suppose  you  have  sailed  28  miles  N.-by- 
W.^W.  Opposite  28  in  the  dist.  column 
under  lJ<4-point  courses  you  find  diff.  lat. 
27.2  miles  and  dep.  6.8  miles. 

(A  tenth  of  a  degree  (or  an  hour)  is  six 
minutes;  a  tenth  of  a  minute,  six  seconds. 
It  is  generally  convenient  to  work  with  these 
decimals.) 

Suppose  you  have  sailed  40  miles  E.-by-N. 
Under  7-point  courses  (reading  from  the  bot- 
tom up)  you  find  opposite  dist.  40,  diff.  lat., 
7.8,  dep.,  39.2. 

Table  2,  Bowditch,  contains  the  same 
elements  worked  for  courses  in  degrees.  You 
should  now  be  prepared  to  work  such  ex- 
amples as  these: 
-  ,.  A  ship  leaving  lat.  36°  15'  N.,  long.  47°  48' 
(J^.,  sails  S.E.-by-E.  78  miles.  Required  the 
diff.  lat.,  the  dep.,  and  the  new  lat. 

Ans.  Diff.  lat.  43.3,  dep.  64.9,  new  lat.  35° 
31'  42"  N. 


y 


44     ELEMENTS   OF   NAVIGATION 

A  ship  leaving  lat.  28°  15'  S.,  long.  43°  18' 
E.,  sails  49  miles  N.W.  What  are  the  diff. 
lat.,  dep.,-and  new  lat.? 

Ans.  Diff.  lat.  34.6  miles,  dep.  34.6,  new 
lat.  27°  40'  24''  S. 

A  ghip  leaving  lat.  1°  10'  N.,  long.  16°  5'  W., 
sails  S.S.E.  168  miles.    Give  same  elements. 

Ans.  Diff.  lat.  155.2  miles,  dep.  64.3  miles, 
new  lat.  1°  25'  12"  S. 

A  ship  leaving  lat.  15°.  15'  N.,  long.  121° 
31'  E.,  steers  63°,  64  miles.  Give  same  ele- 
ments. 

Ans.  Diff.  lat.  29.1,  dep.  57,  new  lat.  15° 
44'  6"  N. 

To  find  the  neiv  longitude. — First  find  the 
diff.  long  by  converting  dep.  into  it.  First 
compute  the  latitude  lying  half-way  between 
that  of  yesterday  and  that  of  to-day.  This 
is  called  the  middle  lat.  Go  to  the  page  in 
Table  2  marked  with  the  number  of  degrees 
of  this  mid.  lat.  which  you  have  just  found, 
and  seek  in  the  diff.  lat.  column  for  the 
amount  of  your  dep.  Opposite  to  it  in  the 
dist.  column  will  be  the  figures  indicating 
the  number  of  minutes  in  the  diff.  long. 

Example:  A  ship  in  lat.  36°  15'  N.,  long. 
52°  18'  W.,  sails  N.E.-by-N.  60  miles;  re- 
quired the  lat.  and  long.  in. 

Table  1,  under  the  head  of  3-]ioint  courses, 
gives  for  60  miles  diff.  lat.  4p.9  miles,  dep. 
33.3.  The  lat.  in  is,  therefore,  37°  4'  54"  N. 
To  find  the  mid.  lat.  add  the  lat.  left  and  the 


DEAD-RECKONING    '  45 

lat.  in,  and  divide  by  2.  Take  the  nearest 
degree  as  your  answer.  In  this  case  the  mid. 
lat.  is  36°  39'  57",  and  as  that  is  nearer  37° 
than  36°  we  take  the  former.  Now  turn  to 
the  page  for  37°  in  Table  2.  Apply  the  dep. 
33.3  in  the  lat.  column;  the  nearest  you  can 
come  to  it  is  33.5,  opposite  which  in  the  dist. 
column  is  42,  which  means  that  in  lat.  37°  a 
dep.  of  33.5  miles  will  equal  42'  diff.  long. 
Long,  left  was  52°  18'  W.  We  have  made  42' 
diff.  long,  to  the  eastward,  thus  diminishing 
our  westerly  longitude.  We  subtract  42' 
from  52°  18'_W.,  and  get  51°  36'  W.  as  our 
long.  in. 

This  process  of  working  out  the  latitude 
and  longitude  is  called  middle  latitude  sailing, 
and  by  it  the  ordinary  problems  of  dead- 
reckoning  are  solved.  The  cases  which  pre- 
sent themselves  in  the  actual  practice  of 
navigation  are  three  in  number. 
^  Case  I. — Course  and  distance  sailed  be- 
ing given,  to  find  the  diff.  lat.  and  dep. 
y  Case  II. — The  lat.  and  long,  left  and  the 
course  and  distance  being  given,  to  find  the 
lat.  and  long.  in. 

Case  III. — The  latitudes  and  longitudes 
of  two  places  being  given,  to  find  the  course 
and  distance  between  them. 

Cases  I.  and  II.  have  been  explained,  ex- 
cept as  to  sailing  true  east  or  west,  which 
is  called  'parallel  sailing.  Here  there  is  no 
diff.  lat.,  and  the  lat.  in  is  the  mid.  lat.    To 


46     ELEMENTS   OF   NAVIGATION 

find  the  diff.  long,  apply  the  distance  sailed, 
which  in  this  case  is  also  the  departure,  in 
the  lat.  column,  and  opposite  it  in  the  dist. 
column  will  stand  the  number  of  minutes  in 
the  diff.  long. 

To  solve  case  III. — Subtract  the  less  lat- 
itude from  the  greater,  and  reduce  the  re- 
mainder to  minutes.  Do  the  same  with 
the  two  longitudes.  Find  the  mid.  lat.  Go 
to  the  page  in  Table  2  marked  with  the 
number  of  degrees  in  the  mid.  lat.,  and  seek 
the  diff.  long,  in  the  dist.  column.  Opposite 
to  it  in  the  lat.  colmnn  will  be  the  dep. 
Now  seek  in  Table  2  for  the  page  where  the 
diff.  lat.  and  the  dep.  stand  beside  one  an- 
other in  their  respective  columns.  The  re- 
quired dist.  will  stand  opposite  in  the  dist. 
column,  and  the  course  either  at  the  top  or 
bottom  of  the  page,  according  as  diff.  lat. 
or  dep.  is  the  greater. 

Remember  that  in  working  mid.  lat.  you 
will  know  either  your  departure  or  your  diff. 
long,  without  looking  in  the  table,  but  not 
both.  The  one  you  need  is  always  opposite 
the  one  you  have. 

Dep.  in  lat.  col.  =  Diff.  long,  in  dist.  col. 
(and  vice  versa).  In  using  Tables  1  and 
2,  if  the  dist.,  lat.,  or  dep.  in  your  problem 
is  larger  than  any  found  in  the  table,  divide 
the  elements  by  10,  because  the  relations  of 
all  the  parts  of  a  right-angled  triangle  one- 
tenth  the  size  of  yours  will  be  just  the  same 


DEAD-RECKONING  47 

if  you  reduce  all  three  sides  to  one-tenth. 
For  instance,  you  have  diff.  lat.  304';  dep. 
2694  miles.  Divide  both  by  10  and  you  have 
30.4  and  269.4,  both  of  which  are  in  the 
tables.  With  those  you  can  find  one-tenth 
of  your  distance,  which  take  out  and  multi- 
ply by  10.  The  angles  all  remain  the  same, 
so  the  course  is  all  right  as  it  stands. 

Example:  A  ship  in  lat.  42°  3'  N.,  long. 
70°  4'  W.,  is  bound  for  St.  Mary's,  lat.  36° 
59'  N.,  long.  25°  10'  W.  What  are  the  course 
and  distance? 


Lat.  left 42°  03'  N.       Long,  left 70°  04'  W. 

Lat.  sought 30°  .59'  N.       Long,  sought 25°  10'  W. 


Diff.  lat 5°  04'  Diff.  long 44°  54' 

Reduced  to  minutes  =  3Q^  Reduced  to  minutes  —  269A 

Middle  lat 39°  31' 


As  the  tables  do  not  run  beyond  300  miles, 
we  take  one-tenth  of  2694  (the  diff.  long.), 
269,  and  under  40°  with  this  number  in  the 
dist.  column  we  get  206.1  dep.  out  of  the" lat. 
column.  Now  we  look  for  a  place  where  the 
diff.  lat.  is  30.4  and  the  dep.  206.1.  As  we  are 
working  with  one-tenth  of  the  dep.,  we  must 
do  the  same  with  304,  the  diff.  lat.,  or,  in  other 
words,  put  a  decimal  mark  before  the  4,  mak- 
ing it  30.4.  We  find  under  the  head  of  114: 
points  diff".  lat.  30.7,  dep.  206.7,  and  opposite 
them  the  dist.  209.  This  is  one-tenth  of  the 
real  distance,  2090  miles.  As  the  diff.  lat. 
was  southward  and  the  diff.  long,  eastward, 
4 


48     ELEMENTS   OF   NAVIGATION 

the  course  must  be  S.  734  points  E.,  E.^S., 
or  98°. 

EXAMPLES 

What  are  the  course  and  distance  from  the 
Cape  of  Good  Hope,  hit.  34°  22'  S.,  long., 
18°  24'  E.,  to  St.  Helena,  lat.  15°  55'  S.,  long. 
5°  45'  W.? 

Ans.    Course,  310°.     Dist.,  1711  miles. 

A  ship  from  lat.  2°  05'  N.,  long.  22°  30'  W., 
sails  W.S.W.  768  miles.  Required  her  new 
lat.  and  long,  and  the  course  and  dist.  to  St. 
Ann's  Island,  lat.  2°  15'  S.,  long.  43°  38'  W. 

Ans.  Lat.  2°  49'  S.,  long.  34°  24'  W. 
Course,  274°.    Dist.,  559.6  miles. 


WORKING    A    TRAVERSE 

If  a  vessel  sailed  24  hrs.  on  one  course,  the 
student  would  not  be  ready  to  compute  her 
dead -reckoning.  But  since  the  course  is 
changed  frequently,  it  is  necessary  to  obtain 
the  component  of  several  courses.  The 
method  of  doing  this  is  called  working  a 
traverse. 

Suppose  a  vessel  to  start  from  Sandy 
Hook  lightship,  lat.  40°  28'  N.,  long.  73°  50' 
W.,  and  sail  in  24  hours  S.E.  7  miles,  E.-by- 
S.  6}/2  miles,  S.W.  9  miles,  and  S.E.-by-S. 
4.35  miles;  where  would  she  be  at  noon  on 


WORKING   A  TRAVERSE 


49 


the  second  day?  The  diagram  shows  us 
that  she  would  be  17.7  miles  about  S.S.E. 
3^E.  of  the  lightship. 

The  method  of  calculating  such  a  compound 
course  or  working  a  traverse  is  as  follows: 
Write  out  the  various  courses  with  their 
1  o     ....        ^~ 


^^> 


TRAVERSE    COURSE    PROM    SM^DT    UOOK    LIGHTSHIP 


50     ELEMENTS   OF   NAVIGATION 

corrections  for  variation,  leeway,  and  devi- 
ation, and  the  distance  run  on  each.  In 
four  columns  h6aded  respectively  N.,  S.,  E., 
W.,  put  down  the  diff.  of  lat.  and  dep.  for 
each  course.  Add  together  all  the  northings, 
all  the  southings,  all  the  eastings,  all  the 
westings.  Subtract  to  find  the  difference  be- 
tween northings  and  southings,  and  you  will 
get  the  whole  diff.  lat.  The  difference  between 
eastings  and  westings  will  give  the  whole  dep. 

With  the  whole  diff.  lat.  and  whole  dep., 
seek  in  Table  2  for  the  page  where  the 
nearest  agreement  of  lat.  and  dep.  with  your 
figures  can  be  found.  The  number  of  de- 
grees at  the  top  or  bottom  of  the  page  (ac- 
cording as  diff.  lat.  or  dep.  is  greater)  will 
give  you  the  course  made  good.  The  distance 
made  good  is  found  in  dist.  column,  oppo- 
site the  agreeing  lat.  and  dep. 

Find  the  lat.  in,  as  already  explained. 

Find  the  long,  in,  as  already  explained. 

Example:  A  ship  in  lat.  31°  15'  N.,  long. 
68°  30'  15"  W.,  sails  by  compass  36  miles 
E.-by-S.,  with  1  pt.  W.  var.,  }4  pt.  E.  dev., 
3^  pt.  port-tack  leeway;  22  miles  S.S.E.  with 
same  variation,  3^  pt.  E.  dev.,  34  pt.  star- 
board-tack leeway;  28  miles  S.-by-E.  with 
same  variation,  3^  W.  dev.,  34  Pt.  port-tack 
leeway;  and  31  miles  S.  with  %  pt.  W.  var., 
3^  pt.  E.  dev.,  and  34  pt-  Port  tack  leeway. 
Required  the  course  and  distance  made  good 
and  the  new  lat.  and  long. 


WORKING  A  TRAVERSE 


51 


Ans.  Course,  145°.  Dist.,  99  miles.  Lat. 
29°  53'  54"  N.,  long.  67°  23'  15"  W. 

In  this  example  there  is  no  subtraction 
of  southing  and  northing,  or  of  easting  and 
westing.  Let  us  suppose  a  case,  however, 
of  a  ship  beating  to  the  eastward,  and  forced 
to  run  off  to  the  northwest  by  some  acci- 
dent. Omitting  the  corrections  of  the  com- 
pass course  for  the  sake  of  brevity,  we  should 
have  a  traverse  like  this: 


Lat.  left,  20°  30'  N.                       Long,  left,  48°  25'  W. 

Course 

Distance 

N. 

S. 

E. 

W. 

8.S.E. 
N.E.J^E. 
S.E.HE. 
W.N.W. 

12 
16 
14 
13 

10.7 
.5.0 

11.1 

8.9 

4.6 
11.9 
10.8 

12.0 

15.7 

20.0 
15.7 

27.3 
12.0 

12  0 

4.3 

15.3 

Course  S.  74°  E.     DLstiincc, 

Lat.  left 26°  30'  00"  N. 

Diff.  lat 4'  18"  S. 

Lat.  in 26°  25'  42"  N. 


10  iiiil(-.s. 

Long,  left 48°  25'  W. 

Diff.  long 17' E. 


Long,  in 48°  08'  W. 


Currents. — In  case  of  a  known  current 
setting  directly  opposite  to  the  ship's  course, 
multiply  the  rate  of  the  current  per  hour 
by  the  number  of  hours  you  are  in  it,  and 
subtract  the  amount  from  the  amount  regis- 
tered by  the  log  on  that  course.  If  the  current 
goes  directly  with  the  ship,  add  the  product 
of  its  rate  by  the  time.    In  case  of  a  current 


52     ELEMENTS   OF   NAVIGATION 

setting  across  the  ship's  course,  enter  the 
direction  of  the  current  in  the  traverse  as  a 
course  and  the  product  of  rate  by  time  as  a 
distance. 

Example:  A  ship  from  lat.  36°  15'  S.,  long. 
101°  14'  E.,  sails  in  24  hrs.  30  miles  N.N.W. 
true  and  68  miles  W.3/^N.  true.  During  12 
hrs.  of  the  day  she  is  in  a  current  setting 
E.3/^S.  at  the  rate  of  2  knots  per  hour. 
Required  her  course  and  distance  made  good. 


Course 

Distanco 

N. 

S. 

E. 

W. 

N.N.W. 

W.i^N. 
E.HS. 

30 
68 
24 

27.7 
6.7 

2.4 

23.9 

11.5 
67.7 

34  4 
2.4 

79.2 
23.9 

32  0 

55.3 

Ans.  Course  made  good,  N.  60°  W.,  or  300"", 
dist.  64  miles. 

HOVE   TO 

A  vessel  hove  to  in  a  gale  comes  up  toward 
the  wind  and  then  falls  off,  and  her  course  is 
a  zigzag.  To  keep  her  reckoning  note  how 
she  heads  when  she  has  come  up  as  far  as 
she  will,  and  again  when  she  has  fallen  off 
to  the  limit.  The  point  half-way  between  is 
to  be  called  the  course.  For  instance,  she 
comes  up  to  east  and  falls  off  to  northeast. 
The  course  is  east-northeast. 


THE  DAY'S  RUN  53 

The  leeway,  variation,  and  de\aation  arc 
applied  to  the  course  thus  ascertained.  Dif- 
ferent ships  make  different  leeway,  and  the 
navigator  must  determine  its  extent  by  care- 
ful observation. 

Every  time  she  begins  to  come  up  she  will  go 
ahead  a  little.  The  speed  of  this  progress  or 
"drift"  is  entered  as  the  rate  in  knots.  The 
rest  of  the  operation  is  the  same  as  in  working 
a  traverse. 

Example:  A  vessel  in  lat.  33°  14'  S.,  long. 
60°  47'  E.,  is  hove  to  on  the  starboard  tack. 
She  comes  up  to  E.-by-S.,  and  falls  off  to 
E.-by-N.;  leeway,  6  points;  drift,  2  knots  per 
hour;  variation,  22°  E. ;  no  deviation;  vessel 
hove  to  24  hours.  What  is  her  position  at 
noon  of  the  second  day? 

Ans.    Lat.  32°  40'  S.,  long.  61°  27'  E. 


THE    DAY  S    RUN 

It  is  customary  to  enter  in  the  log-book 
the  position  of  the  ship  at  noon  by  D.  R. 
as  well  as  by  observation,  and  the  day's  run 
is  therefore  worked  up  in  traverse  form. 
Where  observations  fix  new  positions,  dis- 
agreeing with  the  D.  R.  (either  lat.  or  long, 
or  both)  the  D.  R.  is  disregarded  in  con- 
tinuing the  work  and  begun  again  from  the 
position  obtained  by  observation. 

Example  (worked  for  12  hours  only). — Mar. 


54     ELEMENTS   OF  NAVIGATION 


ttl 

^>' 

^>'    ^»'    &:' 

c 

o 

K* 

'.C  t^      CO  CC  OS      t- 

►J 

Sot-o 

C-l 

OT  >0        PO  CC  -1<        (M 

tOiO'* 

o_o 

o              o    o 

Q 

in 

I-;            ifjCTj 

o 

C 

T 

■^ 

^  Tl" 

■^ 

05N.O 

^ 

oc-oo.-< 

o  ><<-*. 

d 

-■  M      ■   •  ^        ?  2  ' 

■  5    P  i^     ^'     »- 

I 

■-.t>.(N 

■-;     >,•*     ^^<» 

5H 

■^or^i 

:te     -^2       -55 

CO(N 

left. 
on  di 

long, 
ong. 

D.  B 

by  ol 
ong.  1 

D.  R 

_  ■  o       -—        .    .— 

■K 

t~rt((M 

Long 
To  n 

Noor 
Diff. 

Long 
Long 
Diff. 

Long. 

15 

OOr-lO 

M 

^ 

■ON^OQO 

CO  'O  CC  CO 

.^j 

<NCO-*cs 

a 

■-i-hMM 

fc 

•7-">y  1 

IMCOOD 

:?;'f5 

' 

■0^ 

o>         oiro 

tj-r^io 

o            o 

P 

o 

O            TO 

~o           c 

+  +  + 

CO 

.1 

m 

O      O      O 

»0  lO  (N 

C3 

»— 1  I-H   t— t 

;    tf        ;  : 

t* 

1    1    1 

%     ^          ■■ 

1 

H 

S^2 

d 

■a       . 

P..M 
M..  . 

P  M 

o 

NTOCO 

Lat.  k 
To  noi 

Noon 

t.  by  ob 
.  to  4  P. 

D.  R.  4 

.  to  S  P. 

D.  R.  8 

lU 

sgss 

■<  o  0,  0, 

E 

Noon  la 
Diff.  lat 

Lat.  bv 
Diff.  lat 

Lat.  by 

1      1 

SHAPING  THE  COURSE  55 

9,  1918.    At  sea.    8  a.m.  sight,  worked  with 

D.  R.  lat.  30°  15'  N.  gavo  long.  46°  28'  W. 
From  8  a.m.  to  noon  ship  sailed  on  285° 
course.  Var.  15°  W.,  dev.  4°  E.  Pat.  log 
reading,  8  a.m.,  126.5;  at  noon,  185.2.  Noon 
sight  gave  lat.  30°  26'  N.  From  noon  ship 
sailed  on  315i  course,  same  var.,  dev.  7°  E., 
till  4  P.M.,  when  Pat.  log  read,  246.6.  Time 
sight  gave  long.  48°  38'  W.  From  4  to  8 
P.M.  ship  sailed  310°,  var.  12°  W.,    dev.   5° 

E.  Pat.  log  at  8  p.m.,  296.8.  Required  D.  R. 
position  of  ship  at  8  p.m.     (Sec  page  54.) 


shaping  the  course 

Having  ascertained  the  position  of  the 
ship,  it  becomes  necessary  to  ascertain  the 
course  required  to  sail  to  reach  the  port 
of  destinatiori.  This  may  be  done  by  using 
the  chart,  if  the  distance  is  small  and  the 
scale  of  the  chart  large.  If  the  distance  is 
considerable  and  the  scale  of  the  chart 
small,  much  inaccuracy  will  follow.  There- 
fore the  course  is  found  by  mid.  lat.  or  Mer- 
cator's  sailing.  For  mid.  lat.  method  see 
Case  III.  of  dead-reckoning. 

The  course  will  be  correct  only  when  the 
distance  is  small.  For  a  long  course  use 
Mercator's  sailing.  In  crossing  the  equator 
treat  the  parts  of  the  course  N.  and  S.  of  it 
separately. 


56     ELEMENTS    OF   NAVIGATION 

For  Mercator's  sailing  we  use  Table  3. 
This  table  contains  the  meridional  parts  cor- 
responding to  the  increase  of  the  length  of 
degrees  as  we  go  toward  either  pole  on  a 
Mercator's  chart.  Find  required  merid.  parts 
by  applying  degi-ees  at  top  of  table,  minutes 
at  side.  Merid.  parts  for  19°  45'  are  1201.4; 
9°  36',  574.9;  29',  28.8. 

To  determine  course  and  distance. — Find 
the  merid.  parts  of  lat.  in  and  lat.  sought. 
Difference  between  them  is  called  meridional 
diff.  lat.  This  is  used  only  in  finding  the 
course,  by  seeking  in  Table  2  for  page  where 
merid.  diff.  lat.  stands  opposite  the  diff.  long., 
former  in  the  lat.  and  latter  in  dep.  column. 
Course  in  degrees  at  top  or  bottom  of  page. 
Under  this  course  apply  proper  {not  merid.) 
diff.  lat.  in  the  lat.  column,  and  find  the  re- 
quired dist.  opposite  to  it  in  dist.  col. 

Example:  What  are  the  course  and  dis- 
tance from  Sandy  Hook  Lightship,  lat.  40° 
28'  N.,  long.  73°  50'  W.,  to  lat.  30°  51'  N., 
long.  72°  45'  W.? 

Lat.  in 40°  28'       Mer.  parts 2644  .5 

Lat.  sought 39°  51'       Mer.  pans 2596.2 

Proper  diff.  lat 0°  37'       Mer.  diff.  lat 48.3 

Long,  in 73°  50'  W. 

Long,  sought 72°  45'  W. 

Diff.  long 1°  05'   =  65' 

On  the  page  in  Table  2  which  has  37° 
at  the  top  and  53°  at  the  bottom  we  find 


NAVIGATION  BY  OBSERVATION  57 

64.7  and  48.7  opposite  one  another.  This 
is  the  nearest  agreement  to  the  meridional 
diff.  lat.  and  the  diff.  long,  that  we  can  find. 
As  the  48.7  is  in  the  right-hand  column  we 
must  read  the  table  up  from  the  bottom, 
and  this  gives  us  a  course  of  127°  or  S,  53°  E. 
Applying  our  proper  diff.  lat.,  37',  in  the  lat. 
colmnn  we  find  37.3,  opposite  which  is  the 
dist.,  62  miles. 

Courses  obtained  by  computation  are  true, 
and  must  be  corrected  for  variation  and  devi- 
ation. Get  the  variation  from  the  chart  and 
then  for  the  magnetic  course  obtain  the  devi- 
ation from  the  Napier  curve,  explained  under 
Compensation  of  Compasses. 


NAVIGATION   BY   OBSERVATION 

Navigation  by  observation  is  carried  on 
by  measuring  the  altitude  of  the  sun,  the 
moon,  or  a  star,  and  computing  from  this 
and  certain  other  data  the  latitude  or  lon- 
gitude of  the  ship.  The  altitude  of  a  celes- 
tial body  is  expressed  in  terms  of  degrees 
and  minutes,  and  is  that  part  of  90°  con- 
tained between  the  body  and  the  sea  horizon. 

An  observer  standing  at  the  point  G  in 
the  diagram  would  see  the  horizon  at  E 
and  F,  and  the  apparent  sky  stretching  from 
one  side  to  the  other  in  a  semicircle,  or 
rather   hemisphere.      Now    a   circumference 


58     ELEMENTS   OF   NAVIGATION 

of  this  semicircle  is  divided,  like  any  other, 
into  180°.  Supposing  the  sun  to  rise  at  E, 
at  D  it  would  be  30°  high,  at  C  50°,  at  B 
70°,  and  at  A,  immediately  overhead,  90°. 
Going  down  the  other  side  its  altitude  would 
continually  decrease.  From  this  we  learn 
that  the  altitudes  of  celestial  bodies  range 
from  0  to  90°,  for  no  matter  in  which  direc- 
tion we  face  the  horizon  the  arc  of  the  sky 


-50 


from  the  horizon  point  opposite  us  to  the 
zenith,  which  is  the  point  immediately  over- 
head, will  measure  90°. 


THE    SEXTANT 

The  first  element,  then,  required  in  any 
problem  of  navigation  by  observation,  is  the 
angular  altitude  of  the  celestial  body  in  use. 
The  measurement  of  this  altitude  is  made  by 
means  of  the  sextant,  or  an  instrument  of  the 
sextant  family. 


THE  SEXTANT  69 

The  principal  parts  of  the  sextant  are 
shown  in  the  accompanying  sketch. 

The  sHding  hmb  (No.  7)  has  a  clamp  slid- 
ing along  the  arc  (No.  10).  A  screw  passes 
through   this  clamp,    and   by   tightening  it 


SEXTANT 

1.  Mirror.  4.  Shade-glasses.  7.  Sliding  limb. 

2.  Telescope.  5.  Back  Shade-glasses.      8.  Reading-glass. 

3.  Horizon-glass.    6.  Handle.  9.  Tangent  screw. 

10.  Arc. 

the  sliding  limb  is  held  firmly  in  any  posi- 
tion at  which  it  is  placed.  It  can,  however, 
be  further  moved  by  very  small  advances 
by  the  use  of  the  tangent  screw  (No.  9). 

The  instrument  is  held  by  the  handle  (No. 
6)  in  the  right  hand,  with  the  telescope 
toward  the  observer's  eye.  He  must  now 
direct  the  telescope  toward  that  part  of  the 


60     ELEMENTS   OF   NAVIGATION 


sea  which  is  directly  beneath  the  celestial 
object  to  be  observed.  His  line  of  sight  will 
pass  through  the  horizon-glass.  He  now 
moves  the  sliding  limb  until  the  image  of  the 
celestial  body,  reflected  by  the  mirror  (No. 
1)  appears  in  the  horizon-glass.  He  then 
tightens  the  clamp  screw,  described  above, 
and  by  means  of  the  tangent  screw  (No.  9) 
moves  the  sliding  limb  just  a  little  more,  so 
that  the  image  "kisses" 
the  horizon,  whi^h  is 
seen  through  the  trans- 
parent half  of  the  hori- 
zon-glass. If  he  can 
make  the  image  split  on 
the  two  halves  of  the 
glass,  as  in  the  cut,  the 
"contact,"  as  it  is  called, 
will  be  all  the  more  ac- 
curate. He  now  reads 
the  angular  altitude 
from  the  scale  on  the 
arc  of  the  sextant  by  means  of  the  reading- 
glass.  The  measurement  is  shown  by  a  small 
vernier  scale  which  runs  along  the  oblong 
opening  in  the  sliding  limb. 

The  arc  itself  is  divided  into  degrees  and 
sixths  of  a  degree  in  this  manner: 


horizon-glass  with  sun 
"kissing  sea" 


7°  6° 

^1     I     I     I     I     I     I 


THE  SEXTANT 


61 


1 

1   1 

1   1 

T  1    1 

0 

The  vernier  is  divided  similarly,  but  its 
parts  represent  minutes  and  sixths  of  a  min- 
ute.   To  read  the  angle  the  zero  point  on  the 
vernier   is  used  as  a 
^^  ^^     starting-point.      If  it 

exactly  coincides  with 
one  of  the  lines  on  the 
scale  of  the  arc,  that 
line  gives  the  meas- 
urement of  the  angle; 
thus,  in  this  case  the 
angle  is  53/2°,  or  5°  30'. 

If,  however,  you  find  the  zero  point  has 
passed  a  line  of  the  arc,  as  in  the  second  case 
shown,  your  angle  is 

more  than  5°  30',  and  ^o  t^o 

you  must  look  along 
the  vernier  to  the 
loft  till  you  find  the 
point  where  the  lines 
do  coincide.  Then 
add  the  number  of 

minutes  and  sixths  of  a  minute  shown  on  the 
vernier  between  zero  and  the  point  of  coinci- 
dence to  the  number  of  degrees  and  minutes 
shown  on  the  arc  at  the  line  which  the  vernier 
zero  has  passed,  and  the  sum  will  be  the  angle 
measured  by  the  instrument. 

Some  instruments  have  the  arc  cut  to  quar- 
ters of  a  degree,  or  15',  and  a  quadrant  is 
cut  to  thirds  of  a  degree,  the  vernier  showing 
minutes  only.    The  sextant  is  the  instrument 


( 

)   1  1       1  1 

y  '  M 

62     ELEMENTS   OF   NAVIGATION 

most  in  use.  The  student  will  require  some 
practice  before  being  able  to  take  and  read 
an  altitude  of  the  sun,  and  a  great  deal  before 
he  can  do  anything  with  the  stars.  An  hour's 
practice  under  an  old  mariner,  however,  will 
do  him  more  good  than  a  hundred  pages  of 
book  instruction. 

Regulate  the  shade-glasses  to  suit  your 
eye.  Those  at  the  top  of  the  instrument 
affect  the  image  of  the  sun  only,  and  serve 
to  deaden  its  brilliancy.  The  back  shade- 
glasses  are  used  when  the  glare  on  the  water 
is  too  powerful.  You  cannot  get  a  good 
contact  with  your  eyes  dazzled. 


SEXTANT   ADJUSTMENTS 

I.  The  mirror  must  be  perpendicular  to  the 
plane  of  the  instrument.  Set  the  sliding 
limb  at  60°.  Hold  the  sextant  face  up,  with 
the  arc  away  from  you.  Place  the  eye  nearly 
in  the  plane  of  the  instrument  opposite  the 
apex  and  look  into  the  mirror.  If  the  image 
of  the  arc  in  the  mirror  and  the  arc  itself 
show  in  one  unbroken  line,  the  adjustment 
is  correct;  if  the  reflected  image  is  lower,  the 
glass  leans  backward;  if  it  is  higher,  the  glass 
leans  forward.  Straighten  the  glass  by  turn- 
in  the  screws  at  its  back. 

II.  The  horizon-glass  must  be  perpen- 
dicular to  the  plane  of  the  instrument.    Set 


SEXTANT  ADJUSTMENTS        63 

the  zero  of  the  vernier  to  the  zero  of  the 
arc.  Hold  the  sextant  almost  face  upward, 
and  look  through  the  sighting-vane  and  the 
horizon-glass  at  the  horizon.  If  the  hori- 
zon line  and  its  image  (seen  in  the  clean 
and  silvered  parts  of  the  glass)  do  not  coin- 
cide, turn  the  screw  at  the  back  of  the  glass 
till  they  do. 

III.  The  horizon-glass  must  be  parallel  to 
the  mirror.  Set  the  zero  of  the  vernier  to 
the  zero  of  the  arc.  Hold  the  instrument  as 
in  taking  an  observation,  and  look  at  the 
horizon.  If  the  line  and  its  image  in  the 
silvered  part  of  the  horizon-glass  coincide, 
tlu;  adj  ustment  is  correct ;  if  they  do  not  show 
in  an  unbroken  line,  adjust  the  horizon-glass 
by  turning  its  screw. 

IV.  The  line  of  sight  of  the  telescope  must 
be  parallel  to  the  plane  of  the  instrument. 
"Screw  in  the  telescope  containing  the  two 
parallel  wires,  and  see  that  they  are  turned 
until  parallel  with  the  plane  of  the  sextant; 
then  select  two  stars,  at  least  90°  apart,  and 
make  an  exact  contact  at  the  wire  nearest 
the  plane  of  the  instrument,  and  read  the 
measured  angle.  Move  the  sextant  so  as  to 
throw  the  objects  on  the  other  wire,  and  if  the 
contact  is  still  perfect,  the  axis  of  the  tele- 
scope is  in  its  right  situation  and  the  telescope 
adjustment  is  correct.  If  the  images  have 
sei)arated,  it  shows  that  the  object  end  of  the 
telescope  droops   toward   the  plane  of  the 


64     ELEMENTS   OF   NAVIGATION 

sextant,  and  if  the  images  overlap,  it  proves 
that  the  object  end  of  the  telescope  points 
away  from  the  plane  of  the  instrument.  This 
will  be  rectified  by  the  screws  in  the  collar 
of  the  sextant.  A  defect  in  the  telescope 
adjustment  always  makes  angles  too  great" 
(Patterson) . 

INDEX   ERROR 

It  is  better  to  have  the  adjusting  done  by 
a  professional  instrument-maker.  Then  let 
the  sextant  alone.  Error  remaining  after  ad- 
justment is  called  index  error.  It  is  found 
thus:  Set  the  sliding  limb  at  o,  hold  the  in- 
strument perpendicularly,  and  look  at  a  star. 
Move  the  sliding  limb  forward  or  backward 
till  the  star  and  its  image  coincide  in  the 
horizon-glass.  Clamp  the  sliding  limb  and 
read  the  angle,  which  is  the  index  error.  If 
zero  on  the  vernier  is  to  the  left  of  zero  on 
the  arc,  the  index  error  is  to  be  subtracted; 
if  it  is  to  the  right,  the  error  must  be  added. 
By  the  sun. — ^Aim  sextant  right  at  the  sun. 
Make  a  contact  of  sun's  image  with  top  of 
sun,  and  read  angle.  Make  contact  of  image 
with  bottom  of  sun  and  read  angle.  One 
reading  will  be  off  the  arc  and  is  marked  +, 
the  other  on  the  arc,  marked  — .  Half  the 
difference  between  tke  two,  marked  with 
sign  of  greater,  is  the  error.  Index  error  is 
usually  expressed  thus:     I.  E.  1°  15'  — ;  or 


CORRECTING   THE  ALTITUDE   65 

I.  E.  2°  8'  +.     The  horizon  and  its  image 
brought  into  line  can  also  be  used. 


HINTS   ON   TAKING   ALTITUDES 

Learn  to  take  a  single  sight  with  accuracy. 
It  is  a  good  thing  to  take  the  mean  of  three 
or  four  sights  when  working  longitude,  but 
you  cannot  always  do  that. 

Oscillating  the  instrument  from  right  to 
left  and  back,  while  taking  a  sight,  will 
make  the  image  skim  the  horizon  so  that 
you  may  make  sure  of  the  point  vertically 
under  it. 

When  fog  obscures  the  horizon  from  the 
deck,  you  can  sometimes  get  a  new  horizon 
by  lowering  yourself  away  in  a  boat. 

In  rough  weather  try  to  get  the  mean  of 
three  or  four  sights.  You  thus  reduce  the 
amount  of  error  caused  by  the  pitching  of  the 
ship. 

Ascertain  the  index  error  before  taking 
every  altitude  or  set  of  altitudes.  The  error 
is  liable  to  change. 


CORRECTING   THE    ALTITUDE 

Certain  corrections  have  to  be  made  to 
all  altitudes  taken  with  a  sextant.  These 
corrections  are  for  dip  of  the   horizon,  re- 


66     ELEMENTS   OF   NAVIGATION 

fraction,  and  in  the  cases  of  the  sun  and 
moon,  for  semi-diameter. 

The  altitude  used  in  the  computation  of 
the  ship's  position  is  that  of  the  center  of 
the  celestial  body.  As  already  explained, 
the  sextant  gives  the  altitude  of  the  upper 
or  lower  edge. 

For  practical  purposes  we  assume  that 
the  diameter  of  the  sun  equals  32'  of  the 
arc  of  the  sky.  Therefore,  if  you  take  the 
altitude  of  the  lower  edge  you  must  add  16', 
or  half  the  diameter,  to  get  the  altitude  of  the 
center.  If  you  take  the  altitude  of  the  upper 
edge,  as  you  might  have  to  do  in  case  the 
lower  one  was  obscured  by  clouds,  you  must 
subtract  16'.  Stars,  having  no  apparent 
diameter,  do  not  call  for  this  correction. 

Dij)  of  the  horizon  means  an  increase  in 
the  altitude  caused  by  the  elevation  of  the 
eye  above  the  level  of  the  sea.  The  simplest 
illustration  of  this  is  afforded  by  the  accom- 
panying figure.  If  the  eye  is  on  the  level 
of  the  sea  at  A,  it  is  in  the  plane  of  the  hori- 
zon CD,  and  the  angles  EAC  and  EAD  are 
right  angles,  or  90°  each.  If  the  eye  is  ele- 
vated above  A,  say  to  B,  it  is  plain  that  the 
angles  EBC  and  EBD  are  greater  than  right 
angles,  or,  in  other  words,  that  the  observer 
sees  more  than  a  semicircle  of  the  sky  and 
hence  his  measurements  are  too  large.  There- 
fore the  correction  for  dip  is  always  subtracted 
from  the  altitude. 


CORRECTING   THE  ALTITUDE   67 

Dip  is  proportionate  to  the  height  of  the 
eye  above  the  water.  Table  14  gives  the  nec- 
essary corrections.  The  navigator  must  ascer- 
tain the  height  of  his  eye  above  the  ship's 
water-Hne. 

Dip  is  subject  to  variations.  Much  dif- 
ference between  temperature  of  the  air  and 
that  of  the  water  displaces  the  horizon.  In- 
crease in  temperature  increases  the  tabulated 


corrections.  Increase  in  wind  diminishes 
them.  Sea  water  colder  than  air,  horizon 
raised,  dip  decreased;  water  warmer,  horizon 
depressed,  dip  increased. 

Error  decreases  as  height  of  eye  increases. 
When  error  is  likely,  take  alt.  from  highest 
point  available. 

Alt.  can  be  taken  against  a  shore  line 
closer  to  ship  than  sea  horizon  would  be. 
For  this  use  Table  15. 

Refraction  is  the  curving  of  the  rays  of 
light  caused  by  their  entering  the  earth's 


68     ELEMENTS   OF   NAVIGATION 

atmosphere,  which  is  a  denser  medium  than 
the  impalpable  ether  of  the  outer  sky.  The 
effect  of  refraction  is  frequently  seen  when 
an  oar  is  thrust  into  the  water  and  looks  as 
if  it  were  bent. 

Refraction  always  causes  a  celestial  object 
to  appear  higher  than  it  really  is.  This  phe- 
nomenon is  greatest  at  the  horizon  and  dimin- 
ishes toward  the  zenith,  where  it  disappears. 
Table  20  gives  the  corrections  for  mean  re-, 
fraction,  which  are  always  subtracted  from 
the  altitudes.  In  the  higher  altitudes,  select 
the  correction  for  the  nearest  degree. 

Avoid  taking  low  altitudes  (15°  or  less) 
when  the  atmosphere  is  not  perfectly  clear. 
Haziness  increases  refraction.  If  compelled 
to  take  a  low  altitude  when  there  appears  to 
be  more  than  the  normal  amount  of  refrac- 
tion, correct  the  refraction  for  the  height  of 
the  barometer  by  Table  21,  Bowditch. 

Table  46  gives  all  corrections  (except  I. 
E.)  in  one. 

Example:  At  sea,  June  27,  1918,  observed 
meridian  alt.:  O  (this  sign  stands  for  the 
sun;  *  for  a  star)  67°  26'  15";  index  error, 
-H  15';  height  of  eye,  25  ft.  Required,  T.  C. 
A.  (true  central  altitude). 

Obs.  alt.  O    67°  26'  15" 

I.  E +     15'  00" 

67°  41'  15" 
Semi-diam +    16'  00" 

67°  67'  15" 


THE  CHRONOMETER  69- 

Dip -       4'  54" 

67°  52'  21" 
Refraction -  23  .6" 


T.  C.  A 67°  51'  57  A" 

In  actual  sea  practice  work  this  way: 

Obs.  alt 07°  20'  15"       I.  E +   15'  00" 

Correction +     25'  43"       Table  40 +    10'  43" 


T.  C.  A 67°  51'  58"  "  +  25'  43" 

THE    CHRONOMETER 

The  chronometer  is  simply  a  finely  made 
and  adjusted  timepiece  placed  in  a  box  and 
swung  in  gimbals,  as  a  compass  is,  to  prevent 
it  from  being  injured  by  the  motion  of  the 
ship. 

The  care  of  a  chronometer  is  not  essen- 
tially a  part  of  the  science  of  navigation, 
but  in  practice  the  na\dgator  has  to  use  and 
care  for  his  o^vn  chronometers,  and  the  author 
has,  therefore,  in  the  latter  part  of  this  book, 
given  some  suggestions  as  to  the  proper  treat- 
ment of  these  instruments. 

The  purpose  of  the  chronometer  aboard 
ship  is  to  register  Greenwich  time.  English 
and  American  navigators  reckon  their  longi- 
tude east  or  west  from  the  Greenwich  merid- 
ian, and,  as  we  shall  learn  further  on,  the 
computation  of  longitude  consists  in  ascer- 
taining the  difference  between  the  time  at 
Greenwich  and  the  time  at  the  ship. 


70     ELEMENTS   OF   NAVIGATION 

The  secondary  reason  for  carrying  a  chro- 
nometer is  that  the  astronomical  data  con- 
tained in  the  Nautical  Almanac  are  all  given 
for  the  Greenwich  time.    Hence: 

Always  note  the  chronom.  lime  of  an  obser- 
vation. 

Every  chronometer  gains  or  loses  a  little 
time  every  day.  When  in  port  the  instru- 
ment is  taken  to  a  maker,  who  regulates  it 
and  ascertains  its  daily  rate  of  losing  or 
gaining.  On  returning  it  to  the  owner,  the 
maker  furnishes  a  memorandum  stating  that 
on  such  and  such  a  date  the  chronometer 
was  so  many  minutes  and  seconds  faster  or 
slower  than  Greenwich  time,  and  was  losing 
or  gaining  so  much  a  day. 

The  navigator,  therefore,  must  correct  the 
time  shown  by  the  chronometer  by  adding 
or  subtracting  the  original  error  -{-  daily 
gain  or  loss,  thus: 

Example:  Chronometer  on  Oct.  11  showing 
2  h.  15  m.  27  s.,  was  3  m.  20  s.  slow  of  Green- 
wich mean  time  (G.  M.  T.)  on  Oct.  1,  and  its 
daily  rate  is  0.8  sec.  losing.  What  is  the 
G.  M.  T.? 

Ans.  Oct.  1  to  11  =  10  days;  0.8  X  10  =  8 
sec.  loss.  On  Oct.  11  chronom.  is  3  m. 
20  +  8  s.  slow. 

Chronom.  time 2  h.  15  m.  27  s. 

Correction +        3  m.  2S  s. 


G.  M.  T 2  b.  18  ro,  55  s. 


THE  NAUTICAL  ALMANAC       71 

Always  make  the  chronom.  correction, 
otherwise  dangerous  error  may  ensue. 

In  practice  a  hack  watch  is  used  in  taking 
sights,  its  time  having  been  compared  with 
that  of  the  chronom.  The  difference  is  des- 
ignated C.-W.,  and  is  easily  applied. 


THE  NAUTICAL  ALMANAC 

The  Nautical  Almanac  is  a  compendium 
of  data  computed  by  the  government  as- 
tronomers for  the  solution  of  the  problems  of 
navigation  by  observation. 

The  first  contents  demanding  the  student's 
attention  are  the  pages  containing  the  dec- 
lination of  the  sun  and  the  equation  of 
time  for  every  two  hours  of  each  day  in  the 
year. 

Declination  is  celestial  latitude.  For  as- 
tronomical purposes  we  locate  an  equator 
in  the  heavens  directly  above  the  earth's 
equator.  The  celestial  north  pole  is  right 
over  our  own.  From  equator  to  either  pole, 
whether  terrestrial  or  celestial,  is  90°.  A 
celestial  body  with  a  dec.  of  20°  is  20°  N.  of 
the  equator.  A  star  directly  over  the  head 
of  a  man  standing  on  the  deck  of  the  Sandy 
Hook  Lightship,  lat.  40°  28'  N.,  would  have 
a  dec.  of  40°  28'  N. 

Because  the  axis  of  the  earth  is  at  an  angle 
with  its  orbit  around  the  sun,  the  sun  ap- 


72    ELEMENTS   OF   NAVIGATION 

parently  moves  to  the  north  in  summer  and 
south  in  winter. 

The  extreme  Umits  of  the  sun's  decHna- 
tion  are  23°  27'  30"  north  and  south.  The 
former  point  is  reached  June  21 — ^our  longest 
day — and  the  latter  Dec,  21.  Half-way  be- 
tween these  dates  the  sun  crosses  the  equator 
going  south.  Hence,  from  June  21  to  Sept. 
21,  the  sun's  dec.  is  N.  and  always  decreasing. 
From  Sept.  21  (or  22)  till  Dec.  21,  the  dec. 
is  S.  and  increasing.  From  Dec.  21  to  Mar. 
21  the  south  dec.  decreases,  and  from  some 
time  on  Mar.  21  the  sun  has  N.  dec,  in- 
creasing. Remembering  these  points,  you 
can  never  be  in  doubt  as  to  whether  dec. 
is  N.  or  S. 

To  ascertain  the  dec.  for  any  time  and 
at  what  rate  it  changes — an  important  point 
— the  navigator  consults  the  Nautical  Alma- 
nac (N.  A.).  Here  is  a  reproduction  of  part 
of  a  page,  giving  the  data  for  the  dec.  and 
the  equation  of  time  (to  be  explained  later). 


Facsimile  of  Part  of  Nautical  Almanac 
October,  1918 

G.  M.  T. 

Declination 

Equation 
of  Time 

h. 
0 
2 

4 
6 

* 

H.D. 

Thursday 

-  9°    3'. 5 
9      5  .4 
9      7  .2 
9      9  .0 

*      *    * 

0   .9 

,  17. 

m.       8. 
+   14     27.0 
14     28.0 
14     29.1 
14     30.1 
*       *     * 

0.5 

THE  NAUTICAL  ALMANAC      73 

The  —  sign  before  dec.  means  that  it  is 
south;  a  +  means  north.  Simihir  signs  before 
the  equation,  however,  mean  add  or  subtract 
in  applying  to  mean  time.  The  dec.  is  given 
for  every  two  hours.  H.  D.  signifies  hourly 
difference  and  applies  to  the  figures  immedi- 
ately opposite  to  it.  If  you  take  an  obser- 
vation at  a  time  not  given  in  the  N.  A,,  select 
the  nearest  hour  and  apply  the  given  H.  D. 
or  fraction  thereof.  If  the  given  hour,  let 
us  say,  is  4  p.m.,  and  the  time  of  your  ob- 
servation is  4  h.  30  m.,  you  will  need  to  add 
half  of  the  H.  D.  to  the  given  dec.  if  the  dec. 
is  increasing,  and  subtract  if  the  dec.  is 
decreasing.  ISTote  carefully  whether  it  is  in- 
creasing or  decreasing.  Or  keep  in  mind  the 
process  of  increase  and  decrease  as  above 
described. 

In  consulting  the  column  marked  G.  M. 
T.,  note  that  the  almanac  records  astronom- 
ical time,  which  begins  at  7ioon  and  is  counted 
through  24  hrs.,  coinciding  with  the  day  of 
a  ship's  run  at  sea.  Thus  10  a.m.  of  June  12 
is  22  o'clock  of  June  11  in  the  N.  A. 

Example:  At  sea,  Oct.  4,  1918.  Chronom. 
time  (corrected),  3  h.  15  m.  15  s.  p.m.  Re- 
quired corrected  dec. 

Dec.  2p.m 4»  10.2'     a       H.  D 1' 

Correction +     1.25'  Time  after  2 1.25  h. 


Cor.  dec 4°  11 .45'  S.       Cor 1 .25' 

or 4"  11'28"S. 


74     ELEMENTS   OF   NAVIGATION 

APPARENT    AND    MEAN    TIME — THE    EQUATION 

Apparent  time  is  that  shown  by  the  sun. 
Mean  time  is  that  shown  by  the  clock. 

The  equation  of  time  is  the  difference 
between  them. 

The  earth  revolves  on  its  axis  once  in  24 
hours,  and  theoretically  the  sun  crosses  the 
meridian  of  any  given  place  at  precisely 
12  o'clock  each  day,  and  it  is  then  noon.  As 
a  matter  of  fact  this  is  not  so.  The  earth 
does  not  revolve  at  a  uniform  rate  of  speed, 
and  consequently  sometimes  the  sun  is  a 
little  ahead  of  time  and  again  it  is  behind. 

Now  you  cannot  manufacture  a  clock  which 
will  run  that  way.  Its  hours  must  all  be  of 
exactly  the  same  length,  and  it  must  make 
noon  at  precisely  12  o'clock  every  day. 
Hence  we  distinguish  clock  time  from  sun 
time  by  calling  the  former  mean  (or  average) 
time  and  the  latter  apparent. 

Your  chronometer  shows  G.  M.  T.  (Green- 
wich mean  time). 

Your  cabin  clock  should  show  L.  M.  T. 
(local  mean  time). 

The  sun  always  gives  L.  A.  T.  (local  ap- 
parent time.) 

Hence,  if  you  wish  to  add  sun  time,  as 
ascertained  from  an  observation,  to  G.  M. 
T.,  you  must  convert  the  former,  L.  A.  T., 
into  L.  M,  T.  by  applying  the  equation  of 
time, 


APPARENT  AND   MEAN  TIME    75 

In  some  operations  you  must  convert  G. 
M.  T.  into  G.  A.  T.,  which  is  also  done  by 
applying  the  equation. 

The  equation  is  given  in  the  N.  A.  with 
a  sign  prefixed,  showing  whether  it  is  to  be 
added  or  subtracted.  Since  the  figures 
given  in  the  time  column  (col.  1)  are  for 
Q.  M.  T.,  these  signs  +  or  —  show  whether 
the  equation  is  to  be  added  to  or  subtracted 
from  G.  M.  T.  to  convert  it  into  G.  A.  T. 
If  you  already  have  ascertained  apparent 
time  and  wish  to  convert  it  into  mean  time, 
obviously  you  must  reverse  the  adding  or 
subtracting  process. 

Always  pick  out  the  equation  for  Green- 
wich, not  local  time. 

The  equation  is  subject,  like  declination, 
to  hourly  variation.  This  is  given  at  the 
foot  of  the  column  of  equations  for  each  day. 
The  equation  itself,  like  the  declination,  is 
shown  for  every  two  hours  of  the  day  at 
Greenwich. 

The  equation  should  be  corrected  just  as 
the  dec.  is,  preferably  by  applying  the  H.  D. 
as  given  in  the  N.  A. 

Do  not  forget  that  the  time  in  N.  A.  begins 
at  noon.  Ten  o'clock  a.m.  of  Jan.  17  is  22 
o'clock  of  Jan.  16. 

An  approximate  knowledge  of  your  longi- 
tude Avill  enable  you  to  determine  whether 
the  chronometer,  which  is  marked  up  to 
12  hrs.  like  an  ordinary  clock,  indicates  a.m. 


76     ELEMENTS   OF   NAVIGATION 

or  P.M.  time  at  Greenwich.,  Turn  the  long, 
into  time  (explained  later).  In  west  long, 
your  time  must  be  earlier  than  Greenwich. 
In  east  long,  vice  versa.  For  example,  in 
round  figures  New  York  is  5  hrs.  west  of  G. 
At  3  P.M.  in  G.  it  is  10  a.m.  in  N.  Y.  In  5 
hrs.  east  long.,  your  clock  showing  3  p.m.,  it 
is  10  A.M.    or  22  hrs.  astronomical  at  G. 


LATITUDE   BY   MERIDIAN   ALTITUDE 

A  meridian  altiiude  is  one  taken  when  the 
celestial  body  observed  bears  true  south  or 
north  of  the  observer,  or  is  precisely  above 
the  meridian  of  longitude  on  which  he  stands. 
In  the  case  of  the  sun  this  is  at  apparent  noon. 

A  meridian  altitude  gives  the  most  ac- 
curate latitude,  for  reasons  which  will  here- 
after be  explained. 

The  general  formula  for  a  meridian  alti- 
tude is  lat.  =  zenith  distance  +  or  —  dec- 
lination. 

Zenith  distance  is  the  distance,  measured 
in  degrees,  from  the  point  precisely  over  the 
observer's  head  to  the  observed  body.  Let 
us  suppose  that  you  and  the  sun  are  both 
north  of  the  equator.  If  now  you  can  ascer- 
tain exactly  how  far  you  are  north  of  the 
sun,  and  how  far  the  sun  is  north  of  the  equa- 
tor, you  will,  by  adding  the  two  measure- 
ments together,  know  your  latitude. 


LAT.   BY   MERID.   ALTITUDE     77 

The  declination  of  the  sun,  obtained  from 
the  N.  A.  and  corrected  for  chronom.  time, 
as  already  explained,  is  the  distance  of  the 
sun  from  the  equator. 

The  zenith  distance  is  the  difference  be- 
tween the  altitude  of  the  sun,  taken  by  the 
sextant,  and  90°.  You  know  that  it  is  90° 
from  the  zenith  to  the  horizon.  Hence,  having 
got  the  altitude  of  the  sun,  you  have  only  to 
subtract  it  from  90°  to  find  how  far  you  are 
from  the  sun.  The  arc  DBC  in  the  diagram 
measures  90°.  If  the  sun  is  at  B,  it  is  48° 
from  C,  the  horizon,  and 
42°  from  D,  the  zenith. 

Now  if  you  are  42° 
north  of  the  sun,  and  it 
is  10°  north  of  the  equa- 
tor, you  must  be  52° 
north  of  the  equator,  or 
in  lat.  52°  N. 

That  is  the  first  and 
simplest  case.  Suppose, 
however,  the  sun  is  in  south  declination,  and 
you  are  somewhere  in  north  latitude.  In  that 
case  your  distance  north  of  the  equator  would 
naturally  be  the  zenith  distance  minus  the 
declination,  because  the  zenith  distance,  alti- 
tude, and  declination  together  would  make  an 
arc  of  over  90°,  and  you  can't  be  over  90° 
north  or  south  of  the  equator. 

Again,  suppose  that  the  sun  is  in  22° 
south  declination,  and  you  are  10°  north  of 


r — -<?• 

>• 

■f. 

\ 

— '>c 

78     ELEMENTS   OF   NAVIGATION 

the  sun.  In  that  case  you  would  have  to 
subtract  the  zenith  distance  from  the  dec- 
lination to  get  your  latitude,  because  the 
sun's  latitude  is  greater  than  yours.  From 
these  considerations  we  deduce  the  following 
rule: 

Begin  to  measure  the  altitude  of  the  sun 
with  the  sextant  a  short  time  before  noon. 
The  altitude  will  constantly  increase  till 
apparent  noon,  when  it  will  stop  and  then 
begin  to  decrease.  You  will  be  able  to  de- 
tect this  by  bringing  down  the  image  of 
the  sun  to  the  horizon  in  the  horizon-glass 
and  carefully  watching  it.  The  highest  alti- 
tude attained  is  the  one  you  need.  At  that 
instant  note  the  chronometer  time. 

To  work  out  the  lat.,  call  the  altitude  S. 
if  the  sun  is  south  of  you,  N.  if  north.  Cor- 
rect the  altitude  for  semi-diam.,  dip,  and 
refraction  as  already  explained.  Subtract 
the  true  central  alt.  from  90°  to  obtain  the 
zenith  dist.  If  the  alt.  is  S.,  name  Z.  D. 
north,  or  vice  versa.  Correct  the  declination 
for  the  chronom.  time  as  already  explained. 
If  Z.  D.  and  dec.  are  both  N.  or  both  S., 
add  them,  and  the  sum  will  be  the  lat.  N.  or 
S.  as  indicated.  If  one  is  N.  and  the  other 
S.,  subtract  the  less  from  the  greater,  and 
the  answer  will  be  the  lat.  named  N.  or  S. 
after  the  greater. 

At  sea,  June  15,  1918.  Obs.  merid.  alt.  O, 
lower  limb,  71°  15',  sun  bearing  S.     Index 


LAT.    BY    MERID.   ALTITUDE     79 

error  —  2';  height  ot  eye,  25  ft.;  chronom. 
3  h.  28  m.  15  s.  p.m.;  chronom.  slow  of  G.  M. 
T.  1  m.  50  s.  on  June  5;  daily  rate,  —  .5  sec. 
Required,  lat.  of  ship. 

Obs.  alt 71°  1.5'  00"  S.         Cor.  Table  46 +    10'  48" 

Correction.  .  .    +       8'  48"  Index  Error —      2'  00" 


T. 

C. 

A...  . 

.  .  71° 

28' 

48" 

S. 

D. 

90° 

00' 

00" 

Z. 

.  .  18° 

3r>' 

12" 

■  N. 

Cor. 

Dec. . . 

.  .  23° 

17' 

57" 

'  N. 

Correction +     8'  48" 

Hourly  diff.  dec.       6" 
Time  after  2  p.m.       1}^  hr. 


Lat 41°  54'  09"  N. 


Cor.  for  dec +9" 

Dec.  2  P.M....   23°  17'  48"  N. 


Cor.  dec 23°  17'  57"  N. 


At  sea,  Sept.  25,  1918.  Obs.  merid.  alt. 
lower  limb  O,  50°  03'  00"  S.  Index  error, 
+  6';  height  of  eye,  20  ft.  Chronom.  2  h. 
15  m.  10  s.  p  M.;  chronom.  slow  of  G.  M.  T. 
on  Sept.  20,  1  m.  10  s.;  daily  rate,  ~  .3  s. 
Required,  lat.  of  ship. 

Obs.  alt 50°  03'  00"  S.        Cor.  Table  46 ...  .    +10'  54" 

Cor +     16'  54"  I.  E +     6' 


T. 

C. 

A..  .. 

,  .  .  50° 

19' 

M" 

■  S. 

D. 

90° 

00' 

00" 

Z. 

, . .  39° 

40' 

00" 

■  N. 

Dec.. 

40' 

27" 

■  S. 

Correction +    16'  54" 


H.  D.  dec 1' 

Time  after  2  p.m.  .  .     .25  hr. 


Lat 38°  59'  39"  N. 


Correction 25'   =   15" 

Dec.  2  p.M 40'   12"  S. 


Cor.  doc 40'  27"  S. 


Taking  meridian  altitudes  is  facilitated  by 
computing  the  approximate  alt.  beforehand 
6 


80     ELEMENTS    OF   NAVIGATION 

and  setting  it  on  the  sextant.  Then  when 
the  body  approaches  the  meridian,  direct  your 
sextant  to  the  horizon  under  the  body  and 
you  will  find  the  image  in  the  horizon-glass. 
You  have  now  only  to  make  the  final  con- 
tact to  get  the  altitude. 

Formula  for  computing  approximate  me- 
ridian alt.  Very  important  to  naval  students 
especially,  as  it  is  daily  in  use.  The  formula 
is  90°  -  L.  +  d  =  h.'(alt.).  Rule:  subtract 
the  lat.  by  D.  R.  from  90°.  Call  remainder 
co-lat.  (complement  of  lat.)  and  name  it  N. 
or  S.  as  lat.  is.  Correct  your  dec.  for  G.  M. 
T.  at  noon  at  ship.  If  co-lat.  and  dec.  are 
of  same  name,  add;  if  of  different  names, 
subtract.    Answer  is  approx.  alt. 

In  actual  practice  at  sea  so  small  a  cor- 
rection to  the  dec.  as  that  in  the  first  ex- 
ample would  be  ignored.  Indeed,  except 
when  nearing  land  or  some  hidden  danger, 
it  is  sufficient  at  sea  to  know  your  lat.  to 
the  nearest  minute.  When  the  chronom.  time 
after  or  before  the  nearest  hour  in  the  N.  A. 
is  small,  and  the  H.  D.  also  small,  correction 
may  be  omitted. 


USE    OF   LAT.    CONSTANT 

The  foregoing  method  is  the  old-established 
way  of  working  a  merid.  alt.,  and  continues 
to  be  used  in  the  merchant  service.     In  the 


USE  OF  LAT.   CONSTANT        81 

navy  it  is  the  rule  to  prepare  a  constant  to 
which  the  obs.  alt.  can  be  appHed  and  the 
lat.  immediately  known.  The  general  for- 
mula for  this  constant  is 

Lat.  =  90°  +  dec.  —  cor.  —  obs.  alt. 

The  correction  is  that  for  the  approx.  alt. 

computed   as   already   explained.     We   now 

have  these  variations  of  the  general  formula: 

I. — Lat.  and  dec.  same  name,  lat.  greater: 

+  90°  +  dec.  -  cor.  -  obs.  alt. 
11. — Lat.  and  dec.  same  name,  dec.  greater: 

-  90°  +  dec.  +  cor.  +  obs.  alt. 
III. — Lat.  and  dec.  opposite  names: 
4-  90°  —  dec.  —  cor.  —  obs.  alt. 
The  constant  is  computed  before  the  observer 
goes  on  deck.     When  the  merid.  alt.  is  ob- 
tained, the  lat.  is  found  by  one  addition  or 
subtraction.     The  first   example  previously 
given  would  be  worked  thus: 

+   90°  00'  00" 

Dec +   23°  17'  57" 

Cor -       8'  48" 


Constant 113°  09'  09" 

Obs.  alt 71°  15'  00" 


Lat 41°  54'  09"  N. 

All  latitudes  by  merid.  alt.  hereinafter  will 
be  worked  by  constant. 

The  correction  in  the  constant  must  be 
treated  algebraically.  It  is  a  quantity  with 
a  sign  -j-  or  — .  If  the  constant  says  + 
cor.,  you  add  it  if  its  sign  is  +,  and  subtract 


.82     ELEMENTS   OF   NAVIGATION 

if  the  sign  is  — .  If  the  constant  says  —  cor., 
you  subtract  it  if  its  sign  is  +  and  add  it  if 
its  sign  is  — . 

LAT.    BY   MERID.    ALT.    OF   A    STAR 

You  can  learn  the  location  of  the  principal 
stars  from  any  good  star  map.  The  N.  A. 
contains  one.  The  best  hours  for  observing 
stars  are  morning  and  evening  twilights,  when 
the  horizon  is  clearly  defined.  Moonlight 
nights  also  bring  out  a  good  horizon.  Ex- 
perienced observers  can  get  star  altitudes  on 
starlight  nights  with  a  first-class  sextant  and 
a  visible  horizon. 

Stars  are  very  serviceable  when  the  sun 
is  invisible  all  day  and  it  clears  at  night. 
Stars  about  to  cross  the  meridian  can  be 
found  very  often;  you  do  not  have  to  wait 
twenty-four  hours. 

The  declinations  of  all  the  stars  available 
for  the  navigator  are  to  be  found  in  the 
back  part  of  the  N.  A.,  in  the  star  tables. 
Those  marked  +  are  N.,  those  —  are  S. 
The  annual  variation  of  declination  is  so 
small  that  the  correction  is  monthly;  heuice 
the  chronometer  time  is  not  taken,  and  no 
allowance  has  to  be  made  for  semi-diameter. 
With  these  exceptions  the  method  of  working 
out  the  lat.  by  a  star's  merid.  alt.  is  the  same 
as  that  for  the  sun. 

Example:    At  sea,  Dec.  7,  1918.    At  10.50 


MERID.  ALT.  OF  A  STAR         83 

P.M.  (L.  M.  T.)  took  merid.  alt.  *  Aldebaran 
(a  Tauris)  75°  21'  S.;  no  index  error;  height 
of  eye,  20  ft. 

+  90°  00'  00" 

Dec +    16°  21'  00' 

Cor +  4'  39" 

Constant 106°  2,5'  39" 

Oba.  alt 75°  21'  00" 

Lat 31°  04'  39" 


Star  corrections  in  Table  46  are  all  minus. 
The  formula  says  subtract  the  correction. 
To  subtract  a  minus  quantity  you  must  add  it. 

It  is  advantageous  to  know  what  star  to 
observe  at  a  given  hour,  or,  in  other  words, 
what  star  is  about  to  cross  your  meridian. 
For  this  you  must  employ  the  right  ascension 
(R.  A.)  of  the  sun  and  the  R.  A.  of  the  re- 
quired star.  The  meaning  of  right  ascension 
will  be  explained  later.  The  R.  A.  of  the 
mean  sun  is  given  for  each  day  and  hour  in 
the  first  pages  of  the  N.  A.  The  R.  A.  of  the 
star  is  in  the  tables  of  places  of  fixed  stars, 
latter  part  of  N,  A. 

Suppose  you  wish  to  learn  what  star  will 
cross  your  meridian  a  little  after  9  p.m. 
Add  9  hrs.  to  the  R.  A.  of  mean  sun.  This 
gives  you  the  R.  A.  of  your  own  meridian. 
If  over  24  hrs.,  subtract  24  hrs.  from  it. 
Select  from  the  star  table  a  star  having  an 
R.  A.  a  little  greater  than  your  own.  That 
will  be  the  next  star  to  cross  your  meridian. 


84     ELEMENTS   OF   NAVIGATION 

Example:  At  sea,  Jan.  5,  1918.  Wished 
to  get  merid.  alt.  of  a  star  about  10  p.m. 

R.  A.  G 18  h.  57  m.  04.6s. 

Time  at  ship 9  h.  30  m.  00 

28  h.  27  m.  04.6  s. 
24  h. 

R.  A.  Merid 4  h.  27  m.  04.6s. 

The  star  table  shows  that  Aldebaran  has 
R.  A.  4  h.  31  m.  15  s.  That  is  the  star  you 
require.  It  will  cross  your  meridian  in  4  m. 
10  s.,  or  the  difference  between  R.  A.  M.  and 
R.  A.  *. 

The  star's  dec.  will  tell  you  whether  to 
seek  it  N.  or  S.  of  you.  If  you  are  in  lat.  N. 
the  star  will  be  S.  of  you  if  its  lat.  is  S.  or 
if  it  is  N.  and  less  than  your  lat.  Lat.  and 
dec.  both  N.,  dec.  greater,  star  N.  of  you. 
Apply  the  same  rule  in  S.  lat. 

If  you  can  get  two  stars,  one  N.  and  one 
S.,  at  about  the  same  time,  the  lat.  will  be 
the  mean  of  the  two  obtained. 


LAT.    BY   MERID.   ALT.    OF   A   PLANET 

The  data  for  planets  follow  those  for  the 
moon  in  the  N.  A.  The  declinations  are 
given  for  each  day,  and  the  daily  variation 
at  the  right  in  small  figures  in  tenths  of 
minutes.  To  save  calculation,  enter  Table 
IV.,  N.  A.,  with  the  daily  variation  at  the 
top  and  the  G.  M.  T.  of  the  observation  at 


MERID.  ALTITUDE  OF  MOON    85 

the  side,  and  take  out  the  number  of  tenths 
of  minutes  to  be  used  as-  correction  for  dec. 
The  rest  of  the  work  is  the  same  as  for  a  star, 
except  that  you  must  note  the  chronom.  time. 
Example:  At  sea,  Feb.  27,  1918.  Obs.  alt. 
Saturn,  75°  21'  S.  No  index  error.  Height  of 
eye,  20  ft.    G.  M.  T.,  10  h.  29  m.  12  s.  p.m. 

+  90°  00'  00" 

Dec +   18°  52'  30" 

Alt.cor...  +      4'  39"  Dec 18''52.1'N. 


Cor.  Table  IV..  .4' 


Const 108°  57'  09" 

Obs.  alt..  .         75°  21'  (M)"  18"  62.5'  N. 


Lat 33°  30'  09"  N. 

LATITUDE    BY   MERIDIAN   ALTITUDE   OF 
rS  THE    MOON 

j  The  moon  is  more  or  less  of  a  nui- 

I  sance,  and  is  not  used  by  expert 
;  navigators  when  it  can  be  avoided. 
!  The  declination  changes  so  rapidly 
i  that  even  minutes  of  time  have  to 
I  be  taken  into  account,  and  one  is 
!  likely  to  be 

j  ^->.  deceived  as 

-""-\J  to  its  semi- 

"'  diameter 

because  of  irradiation,  which 

makes  the  moon  at  times  look 

larger  than  it  is.    Furthermore, 

PARALLAX  in  using  the  moon  we  have  to 


allow  for  parallax. 


86     ELEMENTS   OF   NAVIGATION 

Parallax  is  the  difference  in  the  angular 
altitude  of  a  celestial  body  as  measured  from 
the  surface  or  the  center  of  the  earth.  It 
is  greatest  when  the  body  is  in  the  horizon, 
and  disappears  when  it  is  at  the  zenith.  The 
sun  is  so  far  away  that  its  parallax  never 
exceeds  9".  The  stars  have  practically  none 
at  all  from  the  earth's  surface.  The  moon, 
however,  is  near  enough  to  make  an  allow- 
ance necessary.  On  the  other  hand,  it  is 
often  visible  in  daylight  and  may  be  used 
at  the  same  time  as  the  sun  to  get  combined 
observations  (see  Sumner  method).  Also, 
it  lights  up  the  horizon  at  night,  greatly 
facilitating  the  navigator's  work.  There- 
fore it  should  be  used  when  helpful,  despite 
the  ready  liability  of  the  computations  to 
error.  The  navigator  must  be  especially 
careful  in  moon  work. 

To  work  a  merid.  alt.  of  the  moon,  note 
chronom.  time  of  obs.  and  begin  as  with  a 
planet,  noting  particularly,  however,  that 
the  dec.  is  given,  like  the  sun's,  for  every 
2  hrs.,  with  the  diff.  for  the  same  interval. 
For  correction  use  Table  IV.,  N.  A.,  as  with 
planet. 

Take  out  the  horizontal  parallax  (found 
in  N.  A.  in  column  headed  H.  P.)  and  with  it 
enter  Table  49  (Bowditch)  applying  it  at  the 
top  and  the  obs.  alt.  at  the  side,  and  take  out 
the  correction  (always  -|-)  for  the  obs.  alt. 
This  is  given  for  H.  of  E.  35  ft.     Small  table 


MERID.  ALT.  BELOW  POLE       87 

gives  changes  for  other  heights;  add  or  sub- 
tract as  table  directs. 

These  corrections  for  dec.  and  alt.  are  used 
in  all  moon  observations.  With  the  cor- 
rected dec.  and  alt.  proceed  as  in  merid.  alt. 
of  sun. 

Example:  At  sea,  July  22,  1918.  Obs. 
merid.  alt.  moon,  lower  limb,  59°  06'  40"  N. 
Index  error,  +  2'.  Height  of  eye,  25  ft.  G. 
M.  T.,  11  h.  16  m.  P.M. 

Doc.  10  P.M..   20°  10'  18"  S. 
Cor.  IM  hr..  -9' 


Hor.  parallax. .  . 

58' 

.4 

Cor.  Table  49. .  . 
Cor.  small  table. 
Index  error 

39' 

+  2' 

24" 
54" 

Cor 

Obs.  alt 

+     42' 
59°  00' 

18" 
40" 

T.  C.  A 

59°  4S' 
90°  00' 

58" 
00" 

;  N. 

Z.  D 

Dec 

30°  11' 
20°  01' 

02" 

18" 

s. 
s. 

Cor.  dec 20°  01'  18"  S. 


By  Const.\nt 

+  90°  00'  00" 

Dec 20°  01'   18" 

Cor -     42'  18" 


Const 109°   19'  00" 

Obs.  alt 59°  06'  40" 


Lat 50°  12'  20"  S.        Lat 50°  12'  20"  S. 


MERIDIAN   ALTITUDE    BELOW   THE    POLE 

It  is  frequently  possible  to  get  an  alti- 
tude of  a  star  when  it  is  crossing  the  me- 
ridian below  the  pole.  The  north  pole  of 
the  heavens  is  marked  very  closely  by  the 
pole-star,  which  is  never  more  than  1°  20' 
distant  from  the  pole.  The  stars  in  the 
northern  part  of  the  heavens  apparently  re- 
volve around  the  pole,   as  may  be  plainly 


88    ELEMENTS   OF   NAVIGATION 

seen  in  the  case  of  the  constellation  known 
as  the  "Dipper."  When  the  given  star  is 
directly  under  the  pole  it  is  on  the  merid- 
ian, and  will  give  the  latitude  just  as  cor- 
rectly as  when  directly  above  it. 

When  in  circumpolar  regions  where  the 
sun  does  not  set  for  six  months,  its  merid. 
alt.  below  the  pole  can  be  taken. 

The  altitude  of  the  pole  is  always  equal  to 
your  lat.  If  the  pole  is  directly  over  your 
head  its  altitude  is  90°,  which  is  your  lat. 
If  you  go  south  10°,  the  pole  will  lose  10° 
of  alt.,  being  now  80°  high.  From  this  fact 
we  get  our  formula.  If  we  know  how  far 
below  the  pole  the  observed  body  is,  we  can 
get  the  altitude  of  the  pole  by  adding  the 
alt.  of  the  body  to  its  distance  from  the  pole, 
called  polar  distance.  We  find  the  polar  dis- 
tance by  subtracting  the  dechnation  from  the" 
distance,  which  is  90°,  between  the  equator 
and  the  pole.  Hence  the  formula  P.  D.  -I- 
alt.  =  lat.,  or  by  the  use  of  the  constant: 
+  90°  -  dec.  +  cor.  +  obs.  alt. 

Example:  Jan.  22,  1918.  Obs.  alt.  a  Ursa 
Majoris  (a  of  the  "Dipper")  below  pole, 
8°  15'  N.    H.  of  E.,  10  ft.    No  index  error. 


+  90°  00'  00" 

Dec -   62°  11'  00" 

Cor +    -       9'  28" 


Const 27°  39'  32" 

Obs.  alt 8°  15'  00" 


Lat 35°  54'  32" 


CONVERSION  OF  ARC  TO  TIME  89 


CONVERSION  OF  ARC  TO  TIME,  AND  VICE  VERSA 

Before  proceeding  further  the  student 
should  learn  how  to  convert  longitude  into 
time  and  time  into  longitude.  .The  former 
operation  will  enter  into  most  of  the  cal- 
culations yet  to  come,  and  the  latter  is 
always  part  of  longitude  workings. 

The  conversion  is  based  on  the  fact  that 
the  sun  takes  24  hours  to  pass  around  the 
360°  of  the  earth's  circumference.  Divide 
360  by  24  and  you  get  the  number  of  de- 
grees he  passes  in  one  hour,  viz.,  15°.  Hence 
15°  of  long.  =  1  hour,  and  1°  =  tV  of  1  hour, 
or  4  minutes.  Furthermore,  15'  of  long.  = 
1  minute  of  time,  and  1'  of  long.  =  rs  of  1 
minute  of  time,  or  4  seconds.  Table  7, 
Bowditch,  gives  the  various  equalizations  up 
to  360°,  but  you  should  be  able  to  do  with- 
out it. 

To  convert  time  into  long. — Multiply  the 
hours  by  15  to  get  degrees.  Divide  the  min- 
utes by  4,  and  add  the  quotient  to  the  num- 
ber of  degrees.  If  any  minutes  are  left  over, 
multiply  them  by  15.  Divide  the  seconds  by 
4,  and  add  the  quotient  to  the  minutes. 
Finally,  multiply  the  remaining  seconds 
by  15. 

Exam-pie:  Turn  4  hrs.,  29  min.,  38  sec. 
into  long. 


90     ELEMENTS   OF   NAVIGATION 

4  4)29(7°  4)38(9' 

15  28  36 

60  1  X  15  =  15'  2  X  15  =  30" 

7  9' 

67°  24' 


To  convert  long,  into  time. — Multiply  each 
member  of  the  quantity  by  4  and  divide  by 
60,  adding  any  figures  left  over  to  the  result 
on  the  right. 

Example:    Turn  50°  40'  15"  into  time. 


50°         40'  15" 

4  4  4 

. h.       m.     m.       — ,  s.    s. 

60)200(3  60)  160(2  +  20  =  22  60)60^  +  40  =  41 
180         120  60 

20  m.  40  s.  00 


Ans.  3  hrs.,  22  min.,  41  sec. 

It  is  from  this  convertibility  of  time  into 
degrees  and  parts  of  degrees  (and  vice  versa) 
that  we  get  the  expression  hour-angle. 

Hour-angle  is  the  distance  of  a  body  east 
or  west  of  the  observer's  meridian,  expressed 
either  in  time  or  angle.  Thus  at  11  a.m. 
the  sun's  hour-angle  is  either  1  hour  or  15° 
E.;  at  1.15  p.m.  it  is  either  1  hour  and  15  min. 
or  18°  45'  W. 

In  other  words,  when  the  celestial  body  is 
east  of  you,  its  H.  A.  is  the  time  it  will  take 
the  revolving  of  the  earth  from  west  to  east 
to  bring  the  meridian  on  which  you  are  under 
the  meridian  on  which  the  body  is,  and  the 


CONVERSION   OF  ARC  TO  TIME  91 


distance  which  your  meridian  travels  in  doing 
it  is  measured  on  the  arc  between  the  two 
meridians. 

An  observer  at  O  (zenith  Z)  sees  one  star 
at  N  and  another  at  M.  The  one  at  N 
has  passed  his  zenith  by  25°,  or  1  h.  40  m., 
and  its  H.  A.  is  west.  The  one  at  M  has 
not  yet  reached 
his  Z  by  30°,  or 
2  h.,  and  its  H.  A. 
is  east. 

The  student 
must  master  these 
relations  of  time 
and  arc.  The  con- 
version of  one  into  the  other  is  conveniently 
made  by  Table  7,  Bowditch,  which  is  readily 
understood. 

When  the  longitude  is  known  the  H.  A.  can 
easily  be  computed.  When  the  long,  is  un- 
known the  H.  A.  must  be  found  after  an 
observation  (explained  under  longitude). 
Since  hour-angles  are  measured  by  the  ap- 
parent movements  of  the  celestial  bodies, 
and  not  by  the  clock,  they  are  measured  by 
apparent  time.  Hence  the  rule  for  finding  H. 
A.  of  sun,  when  long,  is  known.  First  find 
the  loc  1  apparent  time,  thus:  at  the  moment 
of  observing  the  body  note  chronom.  time. 
Turn  your  long,  into  time.  If  it  is  E.  add 
it  to  the  G.  M.  T.  If  W.,  subtract.  Result 
is  local  mean  time  (L.  M.  T.).     Apply  the 


92     ELEMENTS   OF   NAVIGATION 

corrected  equation  of  time.     Result  is  local 
apparent  time  (L.  A.  T.). 

In  case  the  sun  is  the  observed  body,  if  the 
L.  A.  T.  is  more  than  12  o'clock  noon  it  is 
the  H.  A.  west.  If  the  local  apparent  time  is 
before  noon,  subtract  it  from  12  hrs.  and  you 
have  the  H.  A.  east.  That  is,  2.30  p.m.  is 
2  h.  30  m.  westerly  H.  A.;  10.30  a.m.  is  1  h. 
30  m.  H.  A.  east. 


LAT.    BY    EX-MERIDIAN   ALT. 

By  using  the  H.  A.  of  the  sun  we  can  get 
our  lat.  when  the  sun  is  not  on  the  merid. 
It  might  be  under  a  cloud  then.  But  if' we 
know  our  long.,  we  can  work  ex-merid.  alts, 
of  the  sun  from  26  min.  before  noon  to  26 
min.  after. 

The  process  rests  on  the  fact  that  near  the 
merid.  the  alt.  varies  as  the  square  of  the 
interval  from  noon.  The  interval  from  noon 
is  the  H.  A.  Table  26,  Bowditch,  gives  the 
change  of  alt.  for  1  min..  Table  27  these  multi- 
plied by  squares  of  intervals. 

Rule:  Take  chronom.  time  of  obs.  Correct 
for  rate.  Find  the  sun's  H.  A.  as  already  ex- 
plained. Enter  Table  26  with  the  dec.  of  the 
sun  at  the  top,  and  the  lat.  by  D.  R.  at  the 
side,  and  take  out  the  change  of  alt.  for  1 
min.  Enter  Table  27  with  the  hour-angle  at 
the  top  and  the  change  of  alt.  at  the  side. 


LAT.   BY   EX-MERID.   ALT.        93 

Pick  out  the  corresponding  reduction  to  the 
merid.,  selecting  units  and  tenths  separately 
and  adding  them.  Add  this  to  the  T.  C.  A. 
to  obtain  the  merid.  alt.  Subtract  this  from 
90°  to  get  Z.  D.,  and  apply  the  dec.  to  get  the 
lat.  as  heretofore  directed. 

Example:  At  sea,  July  11,  1918.  Lat.  by 
D.  R.  50°  10'  00"  N.,  long.  40°  W.  Obs. 
ex-merid.  alt..  O  61°  45'  30".  H.  of  E.,  15 
ft.;  L  E.,  4'  — .  Chronom.  time  (corrected) 
2  hrs.,  38  min.,  00  sec.  p.m 


G.  M.  T..     2h.  38  m. 
Long.  W..     2  h.  40  m. 

00  3. 
00  8. 

P.M. 

Table  26,  var. 

.  of  alt.. 

.  2.5" 

*L.  M.  T.   11  h.  58  m. 
Equation.            —5  m. 

00  8. 
13  8. 

A.M. 

TabI 
Tabl 

Cor. 

Ie27 
le27 

for  alt... 

.   61°  45' 

+  7' 

2" 
.5" 

= 

1'  45" 
25" 

L.  A.  T.  .    11  h.  52  m. 
12  h.  00  m. 

47  s. 

00  8. 

A.M. 

30" 
42" 

;s. 

2'  10" 

H.  A 7  m. 

Obs.  .-ilt 

Cor.  Table  46 .  . 

13  s. 

E. 

T.  C.  A 

.   61°  53' 
+  2' 

12" 
10" 

Merid.  alt 

.   61°  55' 
90°  00' 

22" 
00" 

Z.  D 

Cor.  Dec 

.    28°  04' 
.    22°  10' 

38" 
36" 

'  N. 
N. 

Lat 

.   50°  15' 

14" 

N. 

♦In  this  case  we  have  to  add  12  hrs.  to  G.  M.  T.  in  order  to 
subtract  long. 

In  this  working  the  merid.  alt.  obtained 
is  that  for  the  merid.  on  which  the  ship  is 
at  the  time  of  observation.     The  noon  lat. 


94     ELEMENTS   OF   NAVIGATION 

must  be  computed  by  D.  R.,  according  to 
the  course  and  dist.  to  noon.  In  the  above 
example,  if  the  ship  were  running  at  12  knots 
per  hour,  N.  40°  E.,  she  would  in  7  min.  13 
sec.  make  1.65  miles  and  her  diff.  lat.  would 
be  about  1.3'. 

SIDEREAL   TIME    AND    RIGHT   ASCENSION 

Astronomical  time  is  reckoned  from  noon 
of  one  day  to  noon  of  the  next,  and  hence  the 
astronomical  day  corresponds  to  the  24  hours 
of  a  ship's  run.  The  hours  are  counted  from 
1  to  24,  so  that  4  o'clock  in  the  morning  of 
Oct.  5  is  astronomically  16  o'clock  of  Oct.  4. 

Right  ascension  is  practically  celestial  longi- 
tude. A  place  on  the  earth  is  located  by  its 
latitude  and  longitude;  a  heavenly  body  by  its 
declination  and  right  ascension.  But  R.  A., 
as  it  is  indicated,  is  not  measured  in  degrees 
and  minutes,  nor  is  it  measured  east  and 
west.  It  is  reckoned  in  hours  and  minutes 
all  the  way  around  the  sky  from  west  to  east 
through  24  hours. 

The  celestial  meridian  from  which  this 
celestial  longitude  begins  is  not  that  of  Green- 
wich, but  is  that  passing  through  the  equator 
at  the  point  where  the  sun  crosses  the  line 
in  the  spring. 

When  we  speak  of  a  star  as  having  a  R.  A. 
of  3  hrs.,  42  min.,  15  sec,  we  mean  that  any 
given  spot  on  the  surface  of  the  earth  will 


SID.  TIME  AND  RT.  ASCENSION  95 

occupy  3  hrs.,  42  min.,  15  sec.  in  revolving 
from  the  prime  meridian  of  celestial  long,  to 
the  meridian  of  the  star. 

You  will  meet  with  the  expression  right 
ascension  of  the  meridian.  That  means  the 
R.  A.  of  the  meridian  on  which  you  are,  and 
in  many  stellar  observations  you  need  to 
know  it  in  order  to  compare  it  with  the  R.  A. 
of  the  star. 

It  so  happens  that  the  R.  A.  of  the  meridian 
and  local  sidereal  time  are  the  same  thing. 
Sidereal  time  is  "star"  time,  as  opposed  to 
solar  or  "sun"  time.  The  sidereal  day  con- 
tains 24  hours,  and  begins  when  the  prime 
celestial  meridian  (at  which  celestial  longi- 
tude— or  R.  A. — begins)  is  right  over  the  me- 
ridian on  which  you  are.  It  is  then  what 
may  be  called  sidereal  noon  at  your  position, 
just  as  it  is  solar  noon  when  the  sun  is  on  the 
meridian. 

Refer  to  the  diagram  and  suppose  R  to 
be  the  prime  celestial  meridian,  and  M  your 
meridian.  When  M  is  under  R  sidereal  time 
at  M  begins.  Also  right  ascension  is  meas- 
ured eastward  in  hours  and  minutes  from  R. 
Now  if  M  occupies  2  hrs.,  15  min.,  12  sec.  in 
revolving  with  the  motion  of  the  earth  to 
A,  when  it  arrives  at  A  it  will  be  2  hrs., 
15  min.,  12  sec.  o'clock  sidereal  time  at  M. 
And  that  must  also  be  the  R.  A.  of  the 
meridian  (M.)  because'  R.  A.  is  measured 
from  the  same  point  as  sidereal  time. 
7 


96    ELEMENTS   OF  NAVIGATION 

The  student  must  now  learn  two  things: 
first,  how  to  find  the  sidereal  time  at  Green- 
wich corresponding  to  any  given  hour  of 
mean  time  there,  and  secondly,  how  to  find 
the  sidereal  time  corresponding  to  any  given 
hour  at  his  own  meridian.  It  is  obvious  that 
if  you  can  find  the  former,  you  can  easily  get 


the  latter  by  applying  the  longitude  of  3'our 
meridian  (converted  into  time). 

A  sidereal  day  measures  in  mean  time — 
that  is,  by  a  chronometer  or  ordinary  clock 
— 23  hrs.,  56  min.,  04  sec.  In  other  words, 
every  hour,  minute,  and  second  in  a  side- 
real day  is  a  little  shorter  than  its  counter- 
part in  a  solar  day.  So,  in  turning  mean 
time  into  sidereal  time,  we  have  to  make 
some  allowances.     Table  8,  Bowditch,  gives 


SID.  TIME  AND  RT.  ASCENSION  97 

the  allowances  for  changing  sidereal  to  mean 
time,  and  Table  9  for  changing  mean  to 
sidereal.  Similar  tables  are  to  be  found  in 
the  N.  A. 

The  N.  A.  will  give  you  the  sidereal  time 
at  Greenwich  noon  for  every  day  in  the 
year  (right  ascension  of  mean  sun,  pages 
2  and  3).  Right  ascension  of  the  mean  sun 
is  the  sidereal  time  at  Greenwich  when  the 
mean  time  clock  shows  noon.  Convert  G.  M. 
T.  into  Greenwich  sidereal  time  (G.  S.  T.) 
thus: 

Add  to  G.  M.  T.  the  G.  S.  T.  for  the  pre- 
ceding noon,  and  the  allowances  given  in 
Table  9,  for  the  number  of  hours,  minutes, 
and  seconds  in  the  G.  M.  T.  If  the  sum  is 
more  than  24  hours,  subtract  24  hours  from 
it,  because  at  the  end  of  24  hours  Sid.  T. 
begins  over  again. 

Example:  Required  G.  S.  T.,  Nov.  2, 
1918,  when  the  G.  M.  T.  by  chronom.  (cor- 
rected) was  7  hrs.,  25  min.,  15  sec.  p.m. 

G.  M.  T 7  h.  25  m.  15  s. 

Sid.  T.  at  G.  at  preceding  noon.  ...    14  h.  43  m.  47 .7  s. 
From  table  7  hrs.  25  m.* 1  m.   13  s. 

Sid.  T.  at  G 22  h.  10  m.  15  a. 

*  The  15  sec.  of  G.  M.  T.  arc  disregarded  because  the  allow- 
ance is  only  .041". 

To  find  local  sidereal  time  (L.  S.  T.),  which 
is  also  R.  A.  M.,  simply  apply  your  long,  (in 
time)  to  the  G.  S.  T. 


98     ELEMENTS   OF   NAVIGATION 

Example:  Required  sidereal  time  at  ship 
Aug.  19,  1918,  when  G.  M.  T.  was  11  hrs., 
15  min.,  20  sec.  p.m.    Long.  60°  15'  W. 


G.  M.  T 11  h.  15  m.  20  s. 

G.  S.  T.  preceding  noon.  9  h.  48  m.  06  .2  s. 
Allowance  11  h.  15  m..       1  m.  50.8s. 


G.  S.  T 21  h.  05  m.  17  s. 

Long.  W 4  h.  01  m.  00  s. 


L.  S.  T.  (or  R.  A.  M.) .  17  h.  4  m.  17  s. 


LATITUDE    BY   THE    POLE-STAR 

The  pole-star  (Polaris)  is  1°  20'  distant  from 
the  North  Pole.  It  apparently  revolves 
around  the  pole,  as  other  circumpolar  stars 
do.  As  already  explained,  the  altitude  of  the 
pole  equals  your  lat.  When  Polaris  is  due  E. 
or  W.  of  the  pole  its  alt.  is  your  lat.  When 
it  is  on  the  meridian  above  the  pole,  your  lat. 
=  alt.  —  1°  20'.  At  its  lower  transit,  lat.  = 
alt.  +  1°  20'. 

The  R.  A.  of  M.  advances  from  0  to  21 
hours  in  exactly  the  same  time  that  Polaris 
appears  to  revolve  around  the  pole,  and  the 
astronomers  have  made  a  table  by  which, 
providing  we  know  the  R.  A.  of  M.,  we  can 
make  the  addition  or  subtraction  to  the 
star's  alt.  at  any  hour.    Hence  the  rule: 

Take  the  alt.  and  note  chronom.  time. 
Correct  both  as  usual.  Find  the  local  sidereal 
time  (which  is  also  R.  A.  of  M.)  as  already 


TIME  AZIMUTHS  99 

explained.  With  the  local  sidereal  time  enter 
Table  I.,  in  the  back  part  of  the  N.  A.,  ap- 
plying the  hours  at  the  top  of  the  column 
and  the  minutes  at  the  side.  You  will  thus 
obtain  a  correction  which,  according  to  sign 
prefixed  to  it,  is  to  be  applied  to  the  obs. 
alt.  of  Polaris.  The  result  will  be  the  ap- 
proximate latitude.  The  problem  is  not  re- 
garded as  giving  a  latitude  as  exact  as  that 
obtained  from  a  star's  meridian  passage. 

Example:  At  sea,  Dec.  20,  1918.  Long. 
45°  15'  W.;  obs.  alt.  of  Polaris,  40°  27'  00"; 
no  I.  E.;  H.  of  E.,  20  ft.;  G.  M.  T.,  11  hrs., 
30  min.,  00  sec.  p.m. 

G.  M.  T 11  h.  30  m.  00  s.  p.m. 

S.  T.  noon 17  h.  53  m.  02  .4  s. 

Allowance  113'2  h.-  .  ■ ' 1  m.  53  .3  .s. 

20  h.  24  m.  55.7  8. 
24 

G.  S.  T - oh.  24m.  55.7s. 

Long.  W 3  h.  01  in.  00  8. 

L.  S.  T 2  h.  23  m.  55 .7  s. 

Obs.  alt 40°  27'  00" 

Table  40 -   5'  31" 


T.  C.  A 40°  21'  29" 

Correction -  1°  05'  48" 


Lat 39°  15'  41"  N. 

TIME   AZIMUTHS 

Under  the  head  of  Deviation  the  student 
was  told  how  to  get  the  dev.  by  an  azimuth 


100       ELEMENTS   OF   NAVIGATION 

of  the  sun.  We  now  see  that  the  L.  A.  T. 
needed  in  consulting  the  azimuth  tables  is 
obtained  by  getting  the  G.  M.  T.  of  the  obs., 
appljdng  our  long,  to  it  to  find  L.  M.  T.,  and 
converting  that  into  L.  A.  T.  with  the  equation. 
To  take  an  azimuth  by  moon,  star,  or 
planet,  we  need  the  H.  A.  of  the  body.  This 
is  always  the  diff.  between  the  R.  A.  M.  and 
the  R.  A.  of  the  observed  body,  which  should 
be  made  clear  by  the  following  diagrams. 
These  express  all  the  relations  of  H.  A.  and 
R.  A.  The  semicircle  denotes  the  arch  of  the 
sky  from  horizon  to  horizon. 


H.  A.  West 


O  =  0bser\'er. 

Z  =  Meridian  of  obsers'er. 

L  =  Prime  meridian  of  celestial 

long. 
T  ^  Star. 
LT  =  R.  A.  of  star. 
LZ  =  R.  A.  of  meridian. 
TZ  =  H.  A.  of  star. 


The  H.  A.  being  west,  the  formulae  are: 

LT  =  LZ  -  TZ  (R.  A.  star  =  R.  A.  M.     -  H.  A.  atari 

LZ  =  LT  +  TZ  (R.  A.  M.     =  R.  A.  star  +  H.  A.  star) 

TZ  =  LZ  -  LT  (H.  A.  star  =  R.  A.  M.    -  R.  A.  star) 

H.  A.  East 


The  H.  A.  being  east,  the  formulae: 

LT  =  LZ  +  TZ  (R.  A.  star  =  R.  A.  M.    +  H.  A.  star) 

LZ  =  LT  -  TZ  (R.  A.  M.     =  R.  A.  star  -  H.  A.  star) 

TZ  =  LT  -  LZ  (H.  A.  star  =  R.  A.  star  -  R.  A.  M.) 


LONGITUDE  BY  TIME  SIGHT       101 

Hence,  to  find  a  star's  H,  A.  find  the  L.  S.  T., 
which  is  your  R.  A.  M.  Get  the  star's  R.  A. 
from  the  N.  A.  Subtract  the  less  R.  A.  from 
the  greater.  If  the  star's  R.  A.  is  greater  the 
H.  A.  is  east,  and  vice  versa,  as  above  shown. 
To  get  the  true  azimuth  enter  the  table 
with  this  H.  A.,  but  in  case  of  moon,  star, 
or  planet,  always  apply  the  H.  A.  in  the 
P.M.  part.  If  the  star's  dec.  is  larger  than 
those  in  azimuth  tables,  take  its  alt.  and  use 
Table  V.,  H.  O.  Book,  200. 


LONGITUDE    BY   TIME    SIGHT   OF   SUN 

The  standard  method  of  ascertaining  longi- 
tude by  observation  is  by  a  chronometer  or 
time  sight. 

Since  the  earth  revolves  (apparently) 
around  the  earth  once  in  24  hours,  passing 
through  15°  of  long,  every  hour,  if  we  can 
ascertain  how  many  hours  and  minutes  east 
or  west  of  GreenwicJi  the  sun  is,  and  how 
many  hours  and  minutes  east  or  west  of  the 
sun  we  are,  we  shall  know  our  long.  When 
the  long,  is  not  known,  then  the  problem  is  to 
find  the  local  H.  A.  of  the  sun. 

The  H.  A.  from  Greenwich  we  carry  with 
us  in  the  shape  of  the  chronom.,  which  tells 
us  G.  M.  T.,  and  that,  of  course,  is  simply 
the  H.  A.  of  the  sun  there.  If  we  find  the 
H.  A.  here — at  our  meridian — the  difference 


102       ELEMENTS   OF   NAVIGATION 

between  the  two  will  be  the  number  of  hours, 
minutes,  and  seconds  we  are  east  or  west  of 
the  Greenwich  meridian,  and  this  quantity  is, 
as  we  have  seen,  convertible  into  the  degrees, 
minutes,  and  seconds  of  longitude. 

The  computation  of  the  H.  A.  of  the  sun 
is  a  problem  in  spherical  trigonometry;  but 
the  navigator  has  only  to  know  how  to  use 
the  tables  prepared  by  the  astronomers  and 
to  employ  simple  arithmetic.  The  necessary 
data  are  the  T.  C.  A.,  the  polar  distance  (P. 
D.)  and  the  lat.  At  the  instant  of  getting  the 
altitude  with  the  sextant  note  the  chronom. 
time  accurately  and  make  the  usual  correc- 
tion. Correct  the  dec.  and  equation  of  time 
according  to  G.  M.  T.  Convert  G.  M.  T. 
into  G.  A.  T.  by  applying  the  equation  as 
directed  by  the  N.  A.  You  need  G.  App, 
T.  because  from  your  observation  of  the  sun 
you  get  L.  App.  T.  If  you  prefer,  you  can 
wait  till  you  have  computed  that,  and  then 
convert  it  into  L.  M.  T.  so  as  to  compare  it 
with  G.  M.  T. 

If  you  are  in  N.  lat.  and  the  dec.  is  N., 
or  in  S.  lat.  and  the  dec.  is  S.,  subtract  the 
corrected  dec.  from  90°  to  get  the  polar 
distance.  If  you  are  in  N.  lat.  and  dec. 
is  S.,  or  in  S.  lat.  and  dec.  is  N.,  add  dec. 
to  90°  to  get  P.  D.  The  rule  for  the  rest  of 
the  operation  is  this: 

Add  together  the  P.  D.,  the  lat.,  and  the 
T.  C.  A.    Divide  the  sum  by  2,  and  call  the 


LONGITUDE  SIGHTS 


srs- 
001 
33 


33 


g^ 


'  DO 


r-o 

Mm 


^ 

'0 

05 
1       ° 

+  0° 

M^5 

»4 

0 

to  01 

8| 

0 

poo 
5q  w 

01  0 
^0 

^8 

107 


o   >«'.o 


CO 


o  to  -  -^ 


BB3 


00 

0535 
CO  00 


P» 


+  r 


33 


108       ELEMENTS   OF   NAVIGATION 

Lat.  37°  09'  N.  Venus  E.  of  merid.  Obs. 
alt.  16°  01'.  Index  error,  +  5'.  Height  of 
eye,  45  ft.  (This  sight  was  actually  taken  at 
sea  on  the  given  date.)    (See  table,  page  107). 


SUMNER  S   METHOD 


We  now  come  to  the  method  which,  in  its 
latest  development,  promises  to  supersede  all 
the  older  ones.  Sunuier's  method,  called 
after  the  American  seaman  who  discovered 
its  principle,  rests  upon  certain  fundamental 
truths  of  navigation. 

Wherever  the  sun  is,  it  must  be  perpen- 
dicularly above  some  spot  on  the  surface  of 
the  earth.  Suppose  the  sun  to  be  immedi- 
ately above  the  center  of  the  circle,  S.  Then 
if  a  man  at  A  takes  an 
altitude,  he  will  get  pre- 
cisely the  same  one  as 
men  at  B,  C,  D,  and  E, 
because  they  are  all  at 
equal  distances  from  the 
sun,  and  hence  on  the 
circumference  of  a  cir- 
cle whose  center  is  S. 
Conversely,  if  several 
observers  situated  at  different  parts  of  the 
earth's  surface  take  simultaneous  altitudes, 
and  these  altitudes  are  all  the  same,  then 
these  observers  must  all  be  on  the  circum- 


SUMNER'S  METHOD 


109 


ference  of  a  circle,  and  only  one  circle.  If 
you  moved  one  observer  to  the  circumference 
of  a  larger  circle,  for  instance,  he  would  be 


farther  away  from 
get  a  smaller  alti- 
Now  such  a  cir- 
of  the  earth  would 
large  that  a  small 
ence,  say  20  or  30 
practically  a 
pose  D  to  be  the 
sun  is  vertical,  and 
the  circumference 
around  this  point, 
at  C,  and  from  an 
you  worked  out 
would  find  yourself 
AB,  which  to  all  in- 
is  a  straight  line  at 
true  bearing  of  the 
C,  as  you  may  dis- 
looking  at  it 


the  sun  and  would 
tude. 

cle  on  the  surface 
be  very  large— so 
arc  of  its  circumfer- 
miles,  would  be 
straight  line.  Sup- 
point  over  which  the 
HP  to  be  part  of 
of  a  circle  drawn 
Suppose  you  were 
altitude  of  the  sun 
your  position.  You 
on  the  little  arc 
tents  and  purposes 
right  angles  to  the 
sun  from  the  point 


Ojj  cern  by  simply 
Suppose  now 
we  continue  the  circle  around  D.  Place  an 
observer  at  J,  and  let  him  take  an  altitude  of 
the  sun.  He  will  be  on  the  circumference  of 
the  same  circle,  but  on  the  small  arc  QN, 
which  is  again  practically  a  straight  line  and 


no       ELEMENTS   OF   NAVIGATION 

at  right  angles  to  the  true  bearing  of  the  sun. 
At  S.  he  would  find  himself  on  the  arc  RT — 
again  a  small  straight  line  at  right  angles  to 
the  bearing  of  the  sun. 

If  you  draw  any  other  circle,  and  mark 


points  of  observation,  you  will  get 

similar  results. 

\  Hence:    Any  person  taking   an 

\      altitude  of  a  celestial  body  must 

^     be,  for  all  practical  purposes,  on  a 

straight  line  which  is  at  right  angles  to  the 

true  bearing  of  the  body  observed. 

Such  a  line  is  called  a  Sunmer  line,  or  a  line 
of  position. 

It   must   now   be   perfectly   clear   to   the 


SUMNER'S  METHOD 


111 


student  that  if  the  sun  bears  due  north  or 
south  of  the  observer,  the  resulting  line  of 
position  in.ust  run  east  and  west;  or,  in  other 
words,  it  is  a  parallel  of  lat.  And  that  ex- 
plains why  a  meridian  observation  gives  the 
most  accurate  lat. 

Again,  if  the  sun  bears  due  east  or  west 
the  resulting  line  of  position  7nust  run  north 
and  south ;  or,  in  other  words,  it  is  a  meridian 
of  longitude.  And  that  explains  why  a  prime 
vertical  observation  gives  the  most  accurate 
longitude.  The  observer  at  J  might  be  well 
o'"'er  toward  Q  or  N — in  other  words,  mis- 
taken considerably  as  to  his  latitude — but 
he  would  get  his  longitude  all  right. 

But  in  the  case  of  the  man  at  S,  the  longi- 
tude cannot  be  kno\vn  exactly  unless  the  lat. 
is.  Transfer  the  line  to  a  chart.  We  know 
that  we  are  somewhere  on  that  line  RT.  If 
the  latitude  is  50°  N., 
we  must  be  at  the  point 
where  the  line  crosses 
the  50th  parallel,  which 
is  at  B.  If  the  lat.  is 
55°,  we  must  be  at  C. 
This  shows  how  neces- 
sary the  lat.  is  in  cases 
where  the  observed 
body  does  not  bear  east 
or  west.    On  the  other 

hand,  if  you  wished  to  get  your  lat.  from  the 
line  RT,  you  would  have  to  know  your  long. 


c 

...^ 

^/ 

/ 

/r-' 

112       ELEMENTS    OF   NAVIGATION 

accurately.  If  the  long,  was  70°  W.,  you 
would  know  you  were  at  C,  lat.  55°. 

Hence  we  get  this  operation  from  a  single 
Sumner  line:  Whenever  you  take  a  chro- 
nometer sight  of  the  sun,  or  any  other  heaven- 
ly body,  from  the  L.  A.  T.  or  the  H.  A.  ob- 
tained in  the  computation  get  the  true  bearing 
of  the  body  from  the  azimuth  tables.  Then, 
through  the  position  obtained,  draw  a  Sum- 
ner line  running  at  right  angles  to  the  true 
bearing. 

You  are  absolutely  sure  to  be  somewhere 
on  that  line  at  the  instant  of  observation; 
you  cannot  possibly  be  on  any  other. 

Here  are  some  uses  of  the  single  Sumner 
line.  Suppose  you  are  standing  in  N.W. 
toward  land,  and  your  position  is  not  quite 
certain.  You  take  a  chronom.  sight  and  get 
a  position  from  which  the  sun  bears  S.  7G° 
W.  (256°).  You  draw  the  Sumner  line  AB 
at  right  angles  to  it,  running  N.  14°  W.  (346°). 
The  line,  if  continued,  will  reach  land  at  C, 
and  if  you  sail  on  it  you  will  make  that  point. 

Suppose  that  at  C  there  stood  a  well-known 
lighthouse,  whose  light  was  visible  18  miles 
at  sea  in  clear  weather.  When  that  light 
popped  into  view  over  the  horizon,  you  could 
at  once  verify  your  position  by  taking  its 
bearing,  and  then  sail  in  with  boldness — not 
forgetting  to  use  the  lead. 

But  suppose  you  do  not  wish  to  make  the 
point  C,  which  is  at  the  end  of  your  Sumner 


SUMNER'S  METHOD 


113 


V/- 


line.  Some  20  miles  farther  up  the  coast 
is  a  well-lighted  harbor,  and  you  wish  to 
make  that.  All  you  have  to  do  is  to  draw  a 
second  line  of  bearing,  parallel  to  the  first 
and  ending  at  the  point  you 
wish  to  make.  Measure  the 
distance  at  right  angles  be- 
tween your  two  lines  of 
bearing.  Sail  over  that 
course  and  distance. 
You  will  then  be  on 
the  second  line  of 
bearing,  when  you 
at  once  take  the 
course  346°  true, 
of  the  first  line,  and  you  are 
bound  to  make  your  harbor. 

But  an    accurate   "fix"   can 
be  obtained  only  at  the  inter- 
section of  the  Sumner  lines.     In 
coastwise  work  this  can  often  be 
found   by  crossing  a  Sumner  line 
with  a  bearing  of  some  object  on 
shore,    say    a    high 
mountain. 

At  sea  the   sim- 
plest   form    is    the 
crossing  of  a  Sum-      sUN 
ner    line     obtained  *^ 

from  a  long,  sight  with  one  obtained  by 
merid.  sight,  which  is  a  parallel  of  lat.  Sup- 
pose that  an  observation  at  8  a.m.  gives  you 


114       ELEMENTS    OF   NAVIGATION 

a  Sumner   line   AB,  at  right  angles   to  the 
sun's  bearing.      From  8   a.m.  to  noon  you 


V-' 


make  60  miles  E.N.E.  From  the  a.m.  posi- 
tion at  E  lay  off  the  course  and  distance, 
and  at  the  end  of  this  draw  the  line  CD, 
parallel  to  the  Sumner  line,  AB.     The  point 


SUMNER'S  METHOD 


115 


G,  at  which  your  noon  parallel  of  lat.  cuts 
CD,  will  be  your  noon  position. 

The  old  established  way  of  making  a  noon 
position  is  this:  Take  your  morning  sight 
for  long.,  but  do  not  work  it  out.  Take  your 
noon  sight  for  lat.,  and  then  by  D.  R.  com- 
pute backward  to   the  correct  lat.   at   the 


time  of  the  morning  sight,  and  with  this  lat. 
work  out  the  longitude.  Then  carry  the 
longitude  up  to  noon  by  D.  R.,  and  thus  es- 
tablish the  lat.  and  long,  at  noon. 

The  method  by  a  Sumner  line  and  a  par- 
allel is  far  shorter  and  quite  as  accurate. 
By  it  you  have  found  that  you  are  on  small 
arcs  of  two  different  circles  at  the  same  time. 


116       ELEMENTS   OF   NAVIGATION 

You  can  be  only  at  their  point  of  intersection. 
And  that  is  the  whole  theory  of  the  Sumner 
method. 

To  find  two  intersecting  Sumner  fines, 
neither  of  which  is  a  parafiel  of  lat.,  you  may 
use  successive  observations  of  the  same  body, 
or  simultaneous  observations  of  two  bodies. 
Supposing  the  ship  to  be  stationary,  take  two 
observations  of  sun,  letting  the  bearing  change 
at  least  2  points.  Sumner  lines  drawn  from 
these  observations  will  cross  at  an  angle  of 
2  points  and  you  must  be  at  the  intersection. 

But  the  ship  is  almost  always  going  and 
you  proceed  as  in  the  work  with  a  Sumner 
line  and  the  noon  parallel. 

Having  taken  your  first  sight  and  drawn 
your  Sumner  line,  from  any  point  on  this 
line,  lay  off  the  course  and  distance  made 
up  to  the  time  of  taking  the  second  sight 
and  drawing  the  second  Sumner  line.  At 
the  extremity  of  the  course-line  draw  a  third 
line  parallel  to  the  first  Sumner  line,  and 
prolong  it  till  it  cuts  the  second  Sumner  line. 
The  intersection  of  this  parallel  with  the 
second  Sumner  line  will  be  the  position  of  the 
ship  at  the"  time  of  the  second  observation. 

Refer  to  the  last  previous  diagram.  If 
you  had  been  at  the  point  G  when  you  took 
your  morning  sight  and  had  been  ignorant  of 
your  lat.,  you  would  at  any  rate  have  got 
the  Sumner  line  CD  at  right  angles  to  the 
sun's  bearing  at  that  time  and  would  have 


SUMNER'S   METHOD 


117 


been  somewhere  on  that  Hne.  If  you  had 
remained  at  anchor  till  noon,  you  would 
have  found  your  lat.  and  your  position  G. 
This  explains  why  the  carrying  forward  of  a 
Sumner  line  till  another  is  obtained  to  inter- 
sect it,  gives  a  true  position. 

(See  example  at  top  of  page  118.) 


KJ'ao' 


!0'Zo' 


fO'/o' 


A — Ship's  1st  position.  A  C — 1st  Sumner  line. 

G — -Ship's  2d  position.  D  E — 1st  Sumner  line  carried 

A  B — Course  between  sights.  forward. 

F  E — 2d  Sumner  line. 
E  is  the  intersection  of  D  E  and  F  E  and  the  correct  pos.  of  ship. 

Lat 40°  48'  N. 

Long 71°  02'  W. 

Lat.  by  D.  R.  wag  2Q  m.  too  far  S, 


118       ELEMENTS   OF   NAVIGATION 

Example:  At  sea,  June  1,  1918.  Obs.  alt. 
O  33°  50'  00";  G.  M.  T.,  8  hrs.,  55  min.,  00 
sec.  P.M.;  H.  of  E.,  20  ft.;  no  I.  E.;  lat.,  40° 
17'  N.     (See  tables,  pages  119,  120.) 

T\v  hours  later  took  another  sight,  which 
gave  a  corrected  alt.  of  11°  15',  G.  M.  T., 
10  hrs.,  58  min.  Ship  in  the  meantime  made 
12  miles,  N.  20°  W.    (See  diagram  page  117.) 

In  practice  the  first  sight  would  give  a  cor- 
rect long,  despite  the  lat.  error,  because  the 
sun  is  on  the  P.  V. 

Example:  Sumner  lines  by  simultaneous 
obs.  of  two  stars,  one  east  and  one  west. 
Jan.  25,  1918.  Obs.  alt.  Procyon  32°  44'  E., 
and  Hamal  58°  21'  W.  Lat.  by  D.  R.,  39°  45' 
N.  H.  of  E.  20  ft.  No  I.  E.  G.  M.  T.  first 
obs.,  12  hrs.,  01  min.,  00  sec;   second  obs.. 


LON&. 


LAT. 


3^*46' K 


TRUE  POS. 
lAT.   il' IK  ERROH 


SUMNER'S  METHOD 


119 


a)  01 


■O  o  g. 

of  • 


t-"T3H     HO 

ao  Si" 


4. 
to 

*.o  CO 

CO 
CO 

+   ° 

9°. 

0< 

mmS 

1— Cn 

MO 

CO 

04»c0 

OO 

o  • 


33 


00  >(^ 


Mco 

33 


pt-" 


►-0 
^CO 
MM 


5^ 

3(C 


to  h-i      _2 


ooo 


120      ELEMENTS   OF   NAVIGATION 


<< 

0C5C 

S8 


s 

00 

^ 

-:0 

CO 

i  i  ? 

^ 

5    i 

- 

«J    . 

OCXQ 

2 

00  00 

o 

o^o 

«o 

cc 

OCl 

N 

t^*«^ 

o 

oo 

05 

EE 

s 

ioe<« 

^ 

lOM 

CCM 

8 

?-c^ 

o 

c-.oo 

CC  TT  ■-■ 

I--;-* 

^  + 

_j 

^ 

w 

C 

z^ 

<N 

^'2 

f:^i 

:  ui 

< 

■  o 

0^ 

6 

CU 

V    . 

<:g 

SUMNER'S  METHOD  121 

12  hrs.,  02  min.,  10  sec.  (See  tables,  pages 
122. 123.) 

It  should  be  ob\aous  to  the  student  who 
has  mastered  the  subject  up  to  this  point  that 
all  observations  of  celestial  bodies  give  Sumner 
lines  of  position. 

For  this  reason  the  navigator  should  ac- 
custom himself  to  view  all  his  operations 
as  applications  of  the  fundamental  prin- 
ciples of  the  Sumner  method,  and  accord- 
ingly draw  a  Sumner  line  at  right  angles  to 
the  true  bearing,  as  obtained  from  the  azimuth 
tables  for  the  local  time  of  the  observation, 
the  lat.  of  ship  and  dec.  of  observ^ed  body. 
In  fact,  the  practice  of  navigation  (in  the 
na\'y,  at  any  rate)  now  makes  the  Sumner 
method  its  fundamental  one,  and  treats  all 
else  as  auxiliar}^ 

Comparison  with  the  compass  bearing 
shows  the  error  of  the  compass  at  the  time 
of  the  observation,  and  the  deviation  on 
the  course  is  readily  ascertained. 

The  patent  log  should  be  read  at  the  time 
of  observation,  in  order  that  the  course  and 
distance  made  good  between  that  and  the 
next  observation  may  be  computed.  This  is 
essential  to  the  carrying  forward  of  a  line  of 
position,  as  previously  described. 


122      ELEMENTS   OF   NAVIGATION 


m  an 

K    M 

NOT 
OO 

ss 

E6 

S£ 

(NC<5 

O  ao 

J3J3 

rSJS 

cct- 

•*C0 

COO 
1-1 -^ 


aJCOM 

^ 

OiOOO 

^ 

ssa 

a 

00 

j3j3 

J3J3 

ss 

6m< 


coo 

COOT  ^ 
o  o  o 
(M  to 


§a 


SUMNER'S   METHOD 


123 


ro  s  " 
&""  § 
o'  5  5 


V 

looc 

OClOf) 

loot) 

OS 

g^g^ 

o 

85  ■  •      o  tr 


en 

I     ° 

to 

c.  o 


>2?0 


oto 

wo 

BBS 

OiOii-' 
CCOiO 

tfl    CO 


or 


li^ 

V)lf- 

B- 

trp- 

g 

ton- 
OO 

B 

33 

§ 

OM 
Wit. 

>(^ 

iobi 

? 

a  CD 

crp- 

OO 
uoo 

.3.3 

WO 


124       ELEMENTS   OF   NAVIGATION 


THE   ST.    HILAIRE    METHOD 

This  method,  as  now  appHed  to  the  Sumner 
system,  bids  fair  to  reduce  astronomical  navi- 
gation to  a  single  formula.  The  principle 
upon  which  it  rests  is  this.  At  any  lat.  and 
long,  the  altitude  and  azimuth  of  a  celestial 
body  can  be  calculated  and  a  Sumner  line 
drawn  accordingly.  The  position  may  be 
assumed  and  the  calculation  made  from  it. 
Then  the  actual  altitude  of  the  celestial  body 
can  be  compared  with  the  calculated  one. 
The  difference  between  the  two  will  deter- 
mine the  correct  place  of  the  Sumner  line. 

The  position  by  D.  R.  may  be  used  as  the 
assumed  one,  or  one  may  be  taken  in  its 
neighborhood.  The  observed  alt.  will  be 
higher  or  lower  than  the  calculated  one. 
If  higher,  you  are  nearer  the  sun  than  you 
assumed;  if  lower,  you  are  further  from  it. 
The  distance  by  which  the  observed  and  as- 
sumed positions  disagree  is  equal  to  the  dif- 
ference between  the  two  altitudes  and  is 
called  the  alt.  diff.,  or  intercept. 

The  difference,  expressed  in  minutes,  is  laid 
off  on  the  line  of  the  body's  true  bearing  in 
minutes  of  sea  distance,  or  nautical  miles, 
and  the  assumed  Sumner  line  is  moved  that 
distance  toward  or  from  the  body.  For  it 
should  be  clear  to  the  student  that  the  line 
of  the  object's  bearing  is  an  arc  of  a  great 


THE  ST.  HILAIRE  METHOD       125 

circle  passing  around  the  circumference  of  the 
earth,  and  therefore  its  minutes  are  nautical 
miles. 

It   cannot   matter   whether   the   assumed 
position  is  too  near  the  observed  body  or 


too  far  away.  The  alt.  of  the  observation 
will  correct  the  error  and  give  what  is  called 
the  "position  point,"  and  through  this  the 
correct  line  of  position  is  drawn. 

If  A  be  the  observed  body  and  AM  the 
line  of  its  true  bearing,  you  may  assume  that 
you  are  somewhere  on  BC  or  FG.    If  the  true 


126       ELEMENTS   OF   NAVIGATION 

alt.  locates  you  on  the  line  DE,  you  are  5 
miles  further  from  the  body  than  you  su}> 
posed  when  you  assumed  the  line  BC,  or 
4  miles  nearer  than  you  assumed  on  FG. 

Hydrographic  Office  publication  No.  200 
contains  all  the  tables  needed  for  this  method. 
From  one  you  can  get  j^our  assumed  alt.  and 
from  another  your  azimuth.  The  manner  of 
obtaining  the  calculated  alt.  from  the  table 
is  fully  explained  in  the  book  and  is  too  long 
for  reproduction  here.  Moreover,  the  prac- 
tice of  most  navigators  is  to  make  their  own 
calculation.  The  process  is  simple  and  quick 
and  it  eliminates  the  errors  to  which  the  in- 
terpolations required  in  using  the  table  render 
the  work  liable. 

There  are  three  formulae  for  calculating 
the  alt.,  but  the  most  satisfactory  is  the  cosine- 
haversine  formula.  This  requires  the  use  of 
Table  44,  Bowditch,  to  find  the  cosine,  and 
Table  45,  Bowditch,*  to  find  the  logarithmic 
and  natural  haversines.  The  last  named  stand 
in  the  columns  abreast  of  the  log.  haversines. 
Compute  what  the  L.  A.  T.  will  be  at  the 
moment  of  taking  obs.,  and  take  obs.  at  that 
time.  Also  compute  lat.  and  dec.  for  that 
time.  Or  assume  a  lat.  and  long,  near  those 
by  D.  R.  and  compute  L.  A.  T.  and  dec. 
accordingly.     Then  proceed  as  follows: 

Add  the  log.  haversine  of  the  L.  A.  T.  or 

*  Unless  you  use  H.  O.  Book  200. 


THE  ST.  HILAIRE  METHOD        127 

H.  A.,  log,  cosine  lat.  and  log.  cosine  dec. 
Sum  is  log.  haversine  of  an  arc  used  only  to 
obtain  the  corresponding  natural  haversine. 
Call  the  arc  M.  If  lat.  and  dec.  are  of  same 
name,  find  diff.  and  take  out  its  natural  hav. 
If  lat.  and  dec.  have  different  names,  add 
them  and  take  out  nat.  hav. 

Add  nat.  hav.  M,  and  nat.  hav.  of  sum  or 
diif.  of  lat,  and  dec.  This  gives  nat,  hav. 
of  the  calculated  zenith  dist.,  which  take  out. 
90°  -  Z.  D.  =  calculated  alt. 

The  difference  between  the  calculated  alt. 
and  the  obs,  alt.  is  the  alt.  diff.  or  intercept. 
Obs.  alt.  larger,  you  are  nearer  obs.  body 
than  your  assumed  position,  and  vice  versa. 

Lay  off  the  alt.  diff.  along  the  line  of  the 
obs.  body's  true  bearing,  toward  or  from  it 
as  the  case  requires.  You  can  plot  this  if 
your  chart  permits,  but  it  is  usually  better 
to  compute  the  new  position  point,  and  draw 
the  correct  Sumner  line  through  it.  To  do 
this  regard  the  true  bearing  of  observed  body 
as  a  course  and  alt.  diff.  as  a  dist.  With 
these  enter  the  traverse  tables  and  get  diff. 
lat.  and  dep.  By  mid.  lat.  get  the  diff.  long. 
Apply  diff.  lat.  and  diff.  long,  to  the  assumed 
position  point  and  obtain  those  of  correct 
point. 

In  selecting  the  log.  hav.  of  L.  A.  T.  it  is 
quicker  to  reckon  in  astronomical  time,  thus: 
22  hrs.,  30  min.  L.  A.  T.,  Jan.  21,  rather  than 
1  hr.,  30  min.  H.  A.  east  on  Jan.  22. 
9 


128       ELEMENTS   OF   NAVIGATION 

Example:  At  sea,  Jan.  15,  1918.  Lat.  by 
D.  R.  30°  10'  N.,  long.  45°  15'  W.  T.  C.  A., 
O  17°  41'  00".  Sun  bearing  S.  and  E. 
L.  A.  T.,  8  hi's.,  28  min.,  56  sec.  a.m.  Re- 
quired, alt.  diff.  and  azimuth  to  lay  off  line  of 
position. 

(For  convenience  ascertainment  of  L.  A. 
T.  and  correction  of  alt.  are  omitted.  L.  A. 
T.  can  be  reckoned  as  20  hrs.,  28  min.,  56  sec. 
ast.  time.) 


L.  A.  T 20  h.  28  m.  56: 

Lat 30°   10'  00"  N. 

Dec 21°   14'  54"  S. 


L.  +  D....51°  24'  54' 


Log.  hav 9.29550 

Log.  cos 9.936S0 

Log.  cos 9.96942 


Log.  hav.  M 9.20172 


Calc.  Z.  D 72°   13' 

90°  00' 


Nat.  hav.  M. 
Nat.  hav 


Nat.  hav.. 


.15911 

.18817 


.3472S 


Calc.  alt 

Obs.  alt 

.  .    17°  47' 
..    17°  41' 

Alt.  diff 

6' 

Azimuth,  N.  129°  E. 


To  find  the  lat.  and  lonj^.  of  the  correct 
position  point,  reverse  sun's  bearing.    Then: 


Course 

Dist. 

Diff.  Lat. 

Dop. 

Diff.  Long. 

N.  51°  W. 

or 
309° 

0 

3.8 

4.7 

5' 

Lat 30°   10'  N. 

Diff.  lat 3'  48"  N. 


New  lat. .  .   30°  13'  48"  N. 


Long 45°  15'  W. 

Diff.  long 5'  W. 

New  long 45°  20'  W. 


THE  ST.  HILAIRE  METHOD        129 

Or  for  practical  purposes,  lat.  30°  14'  N., 
long.  45°  20'  W.  Mark  this  point  on  chart 
and  through  it  draw  correct  line  parallel  to 


•t<t°)5'  •**• 


30'/o'  ht. 


A  B.  assuniod  line  of  position  drawn  through  lat.  30°  10'  N.,  long. 
46°   15'  W. 

C  D,  correct  line  of  position  6  miles  away  from  S,  the  Sun,  be- 
cause obs.  alt.  is  smaller  than  calc.  alt. 

assumed  line  as  in  above  diagram.  Where 
simultaneous  sights  of  stars  are  taken  the  same 
method  applies.  Where  successive  sights  with 
intervening  run  are  used,  bring  forward  the 
first  correct  position  line  to  the  second  one, 
as  shown  in  diagram  on  page  130. 

ST.    HILAIRE    MERIDIAN    FORMULA 


As  you  approach  close  to  the  meridian  the 
cosine-haversine  formula  becomes  uncertain, 
because   the   hour-angle   is   so   small.     The 


130       ELEMENTS   OF   NAVIGATION 

author  has  used  it  with  H.  A.  of  6  minutes 
with  good  results.  On  the  meridian  it  cannot 
be  used,  because  the  H.  A.  is  zero.  But  the 
St.  Hilaire  method  of  a  calculated  alt.  is 
nevertheless  available.  You  may  compute 
the  zenith  distance  by  the  formula  P.  D.  — 
co.-lat.  =  Z.  D.,  as  illustrated  in  the  example 
on  page  131,  kindly  sent  to  the  author  by  the 
Hydrographic  Office.  It  will  be  noted  that 
two  erroneous  lats.  are  assumed,  one  north 
and  one  south  of  the  correct  one. 


A — Assumed  position  of  ship  at  first  sight. 
,  B  C — Sumner  line  1st  assumed  pes. 

D  E — Corrected  1st  Sumner  line. 

P  O — Ship's  course  between  sights. 

F  G — Sumner  line  at  second  pos.  by  D.  R. 

H  L — Corrected  2d  Sumner  lino. 

D'  E' — Parallel  to  1st  corrected  Sumner  line,  brought  forward 
to  cut  corrected  second  time. 

M — Intersection  of  H  L  and  D'  E',  ship's  correct  position  at 
time  of  second  sight,  showing  that  assumed  position  on  first  line, 
IJ  C,  was  too  far  north. 


ST.  HILAIRE  MERID.  FORMULA     131 

At  sea,  April  21,  1917,  in  lat.  by  dead- 
reckoning  either  50°  20'  N.,  long.  20°  20'  W., 
or  lat.  50°  10'  N.,  long.  20°  20'  W.,  the  true 
meridian  alt.  was  51°  30',  sun  bearing  south; 
find  the  intercept  and  lat. 


L.  A.  T Oh.     Om.     Os.        Alt .'51''  30' 

Long 1  h.  21  m.  20  s.       Z.  D 38°  .30' 


G.  A.  T 1  h.  21  m.  20  8.       Doc 11°  45.5'  N. 

Equation. ...    —        1  m.   15  ».       Cor -f-       1.1' 


G.  M.  T 1  h.  20  m.  05  s.       Doo 1 1°  40 .0'  N. 

P.  D 78°  13.4 


Co-I>at 39°  40'  Co- Lat .39°  50' 

P.  D 7S°  13  4'  P.  D 78°   13.4' 


Calc.  Z.  D 38°  .33.4'  Calc.  Z.  D 3S°  23  4' 

True  Z.  D. .  .  .  .  .   38°  .30  0'  True  Z.  D .38°  30  0' 


Alt.  diff 3.4' S.  6.6' N. 


Lat.  D.  R .50°  20'  N.  Lat 50°  10'  N. 

Alt.  diff 3  .4'  S.  Alt.  diff 6 .6'  N. 


True  lat 50°  16.6'  N.   True  lat 50°  16.6'  N. 

The  author  has  proposed  a  formula,  which 
seems  to  him  to  be  more  convenient.  This 
formula  has  been  submitted  to  the  Hydro- 
graphic  Office  and  pronounced  correct 

Compute  the  meridian  altitude,  just  as 
you  do  when  ascertaining  the  alt.  correction 
for  a  noon  constant.  Apply  the  alt.  cor.  to 
the  computed  alt.  to  get  the  correct  calc.  alt. 
Compare  the  obs.  alt.  with  it  and  obtain  the 
alt.  diff.  Apply  this  to  the  lat.  assumed  (or 
lat.  by  D.  R.),  making  it  nearer  or  further 


132       ELEMENTS   OF   NAVIGATION 

from  the  sun  as  the  alt.  cliff,  indicates,  and 
you  will  have  the  correct  lat. 

Example:  At  sea,  June  15,  1918.  Obs. 
merid.  alt.  Sun,  71°  15'  00"  S.  Index  error, 
-  3'.  H.  of  E.,  25  ft.  G.  M.  T.,  3  hrs.,  01 
min.,  15  sec.  p.m. 

Old  Way  St.  Hilaire 

Obs.  alt 71°  15'  00"  Lat.  by  D.  R.  .    41°  50'  00"  N. 

Cor +       7'  48"  90°  00'  00" 


T.  C.  A 71°  22'  48"  Co-lat 48°  10'  00"  N. 

90°  00'  00"  Cor.  dec 23°  18'  00"  N. 


Z.  D 18°  37'  12"  N.       Calc.  alt 71°  28'  00" 

Cor.  dec 23°  18'  00"  N.       Cor.  Table  46. .    -       7'  48" 


Lat 41°  55'  12"  N.       Cor.  calc.  alt.. .    71°  20'  12" 

Obs.  alt 71°  15'  00" 


Alt.  diff 5'  12" 

Lat.  by  D.  R...    41°  50'  00" 


Cor.  lat 41°  55'  12"  N. 

It  is  important  to  remember  to  apply  the 
alt.  correction  to  the  computed  alt.  Other- 
wise you  would  have  to  apply  it  to  the  obs. 
alt.  and  so  lose  time.  This  operation  is  pre- 
paring a  constant,  and  if  you  transfer  the 
correction  from  the  obs.  alt.  to  the  constant, 
you  must  change  the  sign.  All  the  work,  except 
finding  the  alt.  diff.  and  applying  it  to  the 
assumed  lat.,  is  done  before  taking  the  ob- 
servation. 

Example:  Mar.  15,  1918.  Obs.  merid.  alt. 
Aldebaran,  71°  21'  00"  S.  No  index  error. 
H.  of  E.,  20  ft. 


ST.  HILAIRE  MERID.  FORMULA     133 

Or-D  Way 

Obs.  alt.^. .  .    7')°  21'  00"  S. 
Correction .  .    —       4'  39" 


T. 

c. 
n. 

A... 

..   75° 
90° 

1(5' 

21" 

Z. 

.  .    14° 

43' 

39" 

■  N. 

Dec. . 

.  .    1(1° 

20' 

4.S" 

'  N. 

Lat 31°  04'  27"  N. 


St.  it 
Lat.  by  D.  R... 

ILAIHE 

30°  r,-,' 

90°  (X)' 

00" 
00" 

;n. 

Co-lat      

r,QO 

05' 
20' 

00" 

48" 

Dec 

l(i° 

Calc.  alt 

Correction 

75° 

+ 

25' 
4' 

48" 
.39" 

;s. 

Cor.  calc.  alt.. . 
Obs.  alt 

75° 
75° 

30' 
21' 

27" 
00" 

s. 

Alt.  diff 

Lat.  D.  R 

30° 

9' 
5.5' 

27" 
00" 

■  N. 

■  N. 

Cor.  Lat 

31° 

04' 

27" 

■  N. 

Tho  samp  worked  with  D.  R.  lat.  in  error 
in  the  opposite  direction. 


Lat.  D.  R 31°  14'  00"  N. 

90°  00'  00" 


Co-lat 58°  46'  00" 

Dec 16°  20'  48" 


Calc.  alt 75°  06'  48' 

Correction +       4'  39'' 


Cor.  calc.  alt 75°   11'  27"  S. 

Obs.  alt 75°  21'  00"  S. 


Alt.  diff 9'  33"  S. 

Lat.  D.  R 31°  14'  00"  N. 


Cor.  lat 31°  04'  27"  N. 

Plotting  ciiarts  on  which  Sumner  lines  can 
be  laid  down  are  available.  Capt.  Fritz 
Uttmark,  of  the  Nautical  Academy,  N.  Y., 
has  devised  a  special  plotting  chart  for  the 
St.  Hilaire  method.  By  it  the  correct  place 
of  the  ship  is  quickly  plotted. 


134       ELEMENTS   OF   NAVIGATION 


GREAT-CIRCLE    SAILING 

It  is  a  peculiar  fact  that  on  a  Mercator's 
chart  a  straight  course  between  two  places 
appears  as  a  curve.  This  is  owing  to  the 
expansion  of  the  degrees  of  lat.  and  long, 
toward  the  poles,  in  order  to  construct  the 
chart  on  the  theory  that  the  earth  is  a  cylin- 
der, as  already  explained.  The  converse  is 
equally  true:  that  a  straight  line  ruled  on 
a  Mercator's  chart  is  really  a  curve  when  you 
come  to  sail  on  it. 

This  is  easily  seen  when  you  draw  the 
two  lines  on  flat  or  spherical  surfaces.  As 
the  meridians  of  longitude  constantly  con- 
verge toward  the  poles,  and  as  courses  are 
all  measured  by  the  angles  they  make  with  the 
meridians,  it  naturally  follows  that  when  you 
draw  the  meridians  all  parallel  to  one  an- 
other, you  must  be  distorting  an  actual  course 
when  you  make  it  cut  all  these  meridians  at 
the  same  angle.  Drawn  on  a  sphere,  your 
straight  course  would  become  a  curve,  known 
as  a  rhumb  line. 

Great-circle  charts  can  be  obtained,  and  on 
them  all  great-circle  tracks  appear  as  straight 
lines.  But  Sir  George  Airy,  Astronomer 
Royal,  designed  the  following  method  of 
drawing  a  great-circle  track  on  a  Mercator's 
chart.  Connect  your  points  of  departure 
and  destination  by  a  straight  line.     Find  its 


GREAT-CIRCLE  SAILING        135 


A 

A 


J^- 

^^ 

^^^^^ 

/ff'////! 

\\\NN^^^ 

^////n 

\\\\Y^ 

/////fH~ 

\  \A  \  \\\\ 

U  1 ' 

A  WW^W 

A 

center  l)y  measurement,  and  draw  a  line  at 
right  angles  and  toward  the  equator.  With 
the  mid.-lat.  between  points  of  departure  and 
destination,  find  "corresponding  parallel"  in 
the  table  on  page  136.  The  perpendicular  line 
must  intersect  this  parallel. 

With  one  point  of  the  dividers  in  this  inter- 
section, with  the  other  point  describe  a  curve 
which  will  pass  through  the  point  of  departure 


136       ELEMENTS    OF   NAVIGATION 

and  that  of  destination.  This  curve  will  be 
the  great-circle  track. 

Blank  spaces  in  table  arise  from  the  fact 
that  in  such  relations  great-circle  sailing  is  of 
no  advantage.  Within  the  tropics,  for  in- 
stance, it  is  of  little  use,  because  the  dis- 
tortion of  the  degrees  on  a  Mercator's  chart 
is  so  small. 

A  ship  on  a  great-circle  track,  except  when 
on  the  equator  or  sailing  N.  or  S.  true,  must 
change  her  course  often  in  order  to  keep  on 
the  track.  Here  the  principle  that  a  small 
arc  of  a  large  circle  on  the  earth's  surface 
is  practically  a  straight  line  may  be  emplo3'ed, 
and  the  successive  courses  laid  off  as  usual 
with  parallel  rules  and  dividers.     You  may 


Middle 

Corresponding  Parallel 

Middle 

Corresponding  Parallel 

Lat. 

opposite  Name  to 

Lat. 

same  Name  as  Lat.  of 

Lat.  of  Places 

Places 

20° 

81°  13' 

* 

* 

22° 

78°  16' 

* 

* 

24° 

74°  59' 

* 

* 

26° 

71°  26' 

* 

* 

28° 

67°  38' 

50° 

4°  00' 

30° 

63°  37' 

60° 

9°  15' 

32° 

59°  25' 

62° 

14°  32' 

34° 

55°  05' 

64° 

19°  50' 

36° 

50°  36' 

66° 

25°  09' 

38° 

46°  00' 

68° 

30°  30' 

40° 

41°  18' 

70° 

35°  52' 

42° 

36°  31' 

72° 

41°  14' 

44° 

31°  38' 

74° 

46°  37' 

46° 

26°  42' 

76° 

52°  01' 

48° 

21°  42' 

78° 

57°  25' 

50° 

16°  39' 

80° 

62°  51' 

52° 

11°  33' 

* 

* 

54° 

6°  24' 

* 

* 

56° 

1°  13' 

* 

* 

GREAT-CIRCLE  SAILING        137 

find  the  distance  on  a  great-circle  course  with 
close  approximation  by  computing  the  lengths 
of  these  short  courses  and  adding  them. 

To  find  the  courses  to  be  sailed,  get  the 
difference  between  the  course  at  starting  and 
that  at  the  middle  of  the  circle,  and  find  how 
many  quarter-points  are  contained  in  it. 
Divide  the  distance  of  half  the  great  circle 
by  this  number  of  quarter-points,  and  that 
will  give  the  number  of  miles  to  sail  on  each 
quarter-point  course. 

Suppose  the  course  at  starting  to  be  N.E., 
and  at  the  center  E.N.E.,  and  the  distance 
from  start  to  center  800  miles.  The  difference 
between  N.E.  and  E.N.E.  is  2  points,  which 
=  8  quarter-points.  Divide  800  by  8,  and 
you  get  100  miles  for  each  quarter-point 
course.  In  other  words,  every  100  miles  you 
change  the  true  course  a  quarter  of  a  point 
easterly. 

Bear  in  mind  that  this  means  true  course. 
Compass  course  must  allow  for  variation 
and  de\'iation. 

Accurate  method  of  measuring  the  distance 
on  a  G.-C.  track. — Turn  the  largest  course 
(always  one  of  the  end  courses)  into  degrees. 
Then  add  the  cosec.  of  the  largest  course, 
cosine  of  the  smallest  lat.,  and  sine  of  the 
diff.  of  long,  between  the  two  places.  Answer 
will  be  sine  of  the  distance  in  degrees  and 
minutes.  As  these  are  degrees  and  minutes 
of  a  great  circle,  which,  like  the  equator,  ex- 


138      ELEMENTS   OF   NAVIGATION 

tends  around  the  full  circumference  of  the 
earth,  multiply  the  degrees  by  60  and  add  the 
minutes,  and  the  result  is  the  distance  re- 
quired. 

If  the  sine  of  the  distance  gives  more  than 
90°,  subtract  the  angle  from  180°,  and  use 
the  sine  of  the  remainder. 


DISTANCE   AND    DANGER   ANGLES 

When  a  light  or  mountain  first  appears 
above  the  horizon,  take  its  compass  bearing 
and  consult  Table  6,  Bowditch,  which  gives 
the  distance  at  which  elevated  objects  can 
be  seen  at  sea.  The  height  of  the  eye  must 
also  be  considered.    Thus: 

At  sea,  running  for  Block  Island  Channel, 
Block  Island  Light,  204  ft.  above  the  level 
of  the  sea,  appeared  above  the  horizon.  Ob- 
server on  bridge  25  ft.  above  sea.  Required 
distance  of  light. 

Table  6 200  ft.   =  18.63  miles'  range  of  visibility. 

6 25  ft.   =     6.59     " 

25.22  miles,  distance  of  light. 

Uncommon  refraction  will  sometimes  make 
a  light  appear  sooner  than  it  ought  to,  and 
the  navigator  must  be  on  the  lookout  for 
such  phenomena.  In  fact,  the  whole  oper- 
ation is  not  to  be  accepted  as  infallible,  for 
at  the  best  it  gives  uncertain  results. 


ANGLES  139 

The  vertical  angle  of  an  object  above  the 
water-line,  measured  by  the  sextant,  may 
also  be  used  to  give  the  distance.  The  navi- 
gator should  possess  Captain  Lecky's  Danger 
Angle  and  Off -Shore  Distance  Tables,  in  which 
are  given  the  sextant  angles  for  heights  up  to 
1,000  ft.  The  vertical  angle  can  be  used  with 
these  tables  when  the  object  is  partly  below 
the  horizon,  or  when  it  is  between  the  horizon 
and  the  observer.  Tables  33  and  34,  Bow- 
ditch,  are  used  in  the  navy.  If  the  object 
is  far  away,  and  the  angle  consequently  very 
small,  it  should  be  measured  both  on  and 
off  the  arc.  For  instance,  with  a  lighthouse, 
first  bring  down  the  center  of  the  lantern 
(just  as  you  would  bring  the  sun)  to  the 
horizon,  and  read  the  angle.  Then  bring  up 
the  horizon  line  to  the  center  of  the  lantern 
by  moving  the  index  bar  of  the  sextant 
toward  you,  and  read  that  angle.  Take  the 
mean  of  the  two,  and  enter  the  tables  under 
the  height  of  the  light.  Opposite  the  sex- 
tant angle  (or  the  nearest  one  to  it)  take  out 
the  distance.  With  a  mountain  bring  down 
the  top  to  the  horizon.  If  the  object  is  be- 
tween you  and  the  horizon,  use  the  object's 
water-line. 

Example:  Oct.  5,  1917,  bound  west,  passing 
Shinnecock  Light,  bearing  N.-by-W.-3^W.  by 
compass,  desired  to  know  distance  of  ship 
from  it.  Vertical  sextant  angle,  from  center  of 
light  to  water-line,  measured  on  and  off,  22'45". 


140      ELEMENTS   OF   NAVIGATION 

In  table  under  160  ft.  and  opposite  22'  50", 
distance  given  is  4  miles. 

Aboard  U.  S.  men-of-war  the  stadimeter 
or  range-finder  may  be  used  to  find  the  dis- 
tance of  any  object  on  shore  not  beyond  its 
limits. 

For  passing  concealed  dangers  the  ver- 
tical sextant  angle  is  used  thus:  Suppose 
that  300  yards  to  the  eastward  of  a  light  45 
ft.  high,  which  you  must  pass  on  the  easterly 
side,  lies  a  shoal  spot  or  a  reef  dangerous  to 
you.  You  therefore  decide  to  pass  300  yards 
outside  of  it,  or  600  yards  from  the  light. 
Under  45  ft.  and  opposite  600  yards  you  find 
the  angle  1°  26'.  You  set  the  sextant  at  that 
angle,  and  watch  for  the  image  of  the  light 
in  the  horizon-glass.  As  long  as  the  angle 
between  the  light  and  the  water-line  is  1°  26' 
or  less,  you  are  600  yards  or  more  from  the 
light,  if  the  angle  becomes  more,  you  are 
inside  of  600  yards.  You  need  not  move  the 
index  bar  at  all,  for  if  the  light  rises  above  the 
water-line  as  seen  in  the  horizon-glass,  the 
angle  is  larger  than  that  set,  and  in  this  case 
that  means  danger;  but  if  it  drops  below,  the 
angle  is  smaller. 

The  horizontal  danger  angle  is  at  times 
extremely  valuable,  and  the  navigator  should 
master  its  use.  It  is  first  necessary  to  learn 
to  take  horizontal  angles  with  the  sextant. 
Hold  the  instrument  face  up.  Look  through 
the  sight-vane  and  horizon-glass  at  the  left- 


ANGLES  141 

hand  object,  and  push  the  index  bar  forward 
till  the  right-hand  object  makes  contact  with 
it.    Then  read  the  angle. 

It  is  a  good  plan  to  take  cross-bearings  this 
way,  noting  the  compass  bearing  of  one  of  the 
objects.  The  bearing  of  the  other  is  at  once 
known  by  the  angle  between  the  two.  If  the 
ship's  head  should  fall  off  between  the  bear- 
ings, and  change  the  deviation,  you  would 
have  only  one  deviation  to  apply. 

The  horizontal  danger  angle  is  used  in 
passing  hidden  dangers.  Suppose  you  wish 
to  pass  at  a  distance  of  a  quarter  of  a  mile 
outside  of  some  hidden  rocks,  and  on  the 
shore  are  certain  objects,  say  a  lighthouse 
and  a  mountain  marked  on  the  chart.  Draw 
a  circle  around  the  rocks  with  a  radius  of 
a  quarter  of  a  mile.  Now  describe  another 
circle  that  will  pass  through  the  lighthouse, 
the  church,  and  the  most  seaward  part  of 
your  first  circle.  From  this  last  point.  A, 
draw  lines  to  the  lighthouse  and  the  church. 
Now  measure  with  a  protractor  the  angle  at 
the  juncture  of  these  two  lines.  Set  that 
angle  (47°  in  the  diagram)  on  the  sextant, 
and  watch  the  selected  objects  with  instru- 
ment face  up.  The  moment  .your  two  objects 
appear  in  the  horizon-glass  you  are  close  to 
your  circle  of  safety,  and  when  they  make 
contact  you  are  on  it.  All  you  have  to  do  is 
to  alter  the  course  of  the  ship  so  as  to  keep 
the  contact,  and  so  sail  around  the  outer 


142      ELEMENTS   OF   NAVIGATION 

part  of  your  circle  till  you  have  rounded  the 
rocks.  If  you  watch  the  angle  closely  this 
cannot  fail,  and  in  narrow  waters  it  is  an  in- 
valuable method. 


In  measuring  vertical  danger  angles  get 
as  close  to  the  water  as  possible,  so  as  to 
remove  error  of  H.  of  E.  But  this  will  in- 
crease your  angle  and  thus  place  you  farther 
away  from  your  danger,  which  is  well,  pro- 
vided there  is  no  other  danger  on  the  other 
side. 

ALLOWANCE    FOR   TIDES 


In  fixing  positions  by  lights,  mountains, 
etc.,  in  passing  over  shoals  and  in  selecting 


ALLOWANCE  FOR  TIDES       143 

anchorage,  remember  that  heights  recorded 
on  charts  are  measured  from  high  water,  or- 
dinary spring  tides,  while  soundings  are  for 
77iean  low  water. 

To  find  the  rise  of  the  tide  or  its  fall. — 
Use  the  following  diagram : 


The  right-hand  side  shows  how  the  tide 
falls  =  3^  of  its  range  for  the  first  hour,  }4 
at  the  end  of  the  second,  ^2  at  the  end  of 
the  third,  and  so  on.  The  left-hand  side 
shows  how  it  rises. 

Remember  that  the  rise  and  fall  do  not 
coincide  with  the  change  of  tidal  current. 
You  must  ascertain  the  duration  of  the  ebb 
and  flow  from  the  published  sailing  directions, 
such  as  the  Atlantic  Coast  Pilot. 

Where  the  range  of  the  tide  is  great,  you 
must  allow  for  it  in  measuring  angular  alti- 
tudes of  shore  marks. 
10 


144       ELEMENTS   OF    NAVIGATION 


RATING   A   CHRONOMETER 

It  is  sometimes  necessary-  on  a  long  voyage 
to  ascertain  the  daily  gain  or  loss  of  the 
chronometer,  owing  to  tlie  fact  that  the  rate 
may  be  affected  by  extremes  of  temperature 
or  other  causes.  The  na\'igator  may  be  far 
away  from  a  maker,  and  hence  must  know 
how  to  ascertain  the  rate  for  himself.  To 
perfonn  the  operation  he  will  require  an  arti- 
ficial horizon.  This  consists  of  a  small  trough, 
which  is  filled  with  absolutely  clean  mer- 
cury, and  covered  with  a  glass  case  which 
permits  the  observer  to  see  the  reflecting 
surface,  and  yet  keeps  wind  and  dust  away 
from  it. 

The  observer  must  now  go  with  his  sex- 
tant, chronometer,  and  artificial  horizon  to 
a  spot  where  the  longitude  is  accurately 
known  to  a  fraction  of  a  second.  This  will 
obviously  be  on  shore,  and  that  is  why  the 
artificial  horizon  must  be  used. 

The  observer  should  station  himself,  sit- 
ting, if  possible,  so  that  the  artificial  hori- 
zon will  ])e  in  a  direct  line  between  himself 
and  the  body  to  be  observed,  and  the  image 
of  the  body  will  be  shown  in  the  mercury. 
Look  through  the  sight-vane  of  the  sextant, 
so  as  to  see  the  image  in  the  mercury  through 
the  horizon-glass.  Bring  down  the  image  re- 
flected by  the  sextant  mirror  till  it  makes 


RATING   A   CHRONOMETER      145 

contact  with  the  image  in  the  mercury.  At 
that  instant  note  the  chronom.  time. 

If  the  obs.  body  is  rising  (east  of  merid.) 
the  two  images  in  the  horizon-glass  will  sep- 
arate, provided  you  are  using  the  lower  limb. 
If  the  body  is  sinking  (west  of  merid.)  they 
will  close. 

The  angle  of  altitude  shown  by  the  sex- 
tant will  be  double  what  it  would  be  with 
a  sea  horizon,  and  must  therefore  be  di- 
\nded  by  2.  The  altitude  is  corrected  as 
usual,  except  for  height  of  the  eye,  which 
does  not  exist  in  this  operation. 

The  remainder  of  the  operation  consists 
of  finding  the  local  mean  time,  and,  by  ap- 
plying the  longitude,  the  correct  G.  M.  T. 
at  the  instant  of  observation.  Thus  the 
error  of  the  chronom.  is  found.  The  ob- 
server now  waits  not  less  than  six  days  (ten 
days  are  better),  and  then  rejjeats  the  process 
at  the  same  place.  From  the  difference  in  the 
error  on  the  two  dates  you  get  the  daily  rate. 

Example:  May  20,  1918.  At  8t.  Anthony 
Point  Light,  Falmouth,  Eng.,  Long.  5°  01' 
W.,  with  artificial  horizon  obtained  alt.  which 
gave  L.  M.  T.  6  hrs.,  50  min.,  08  sec.  Adding 
20  min.,  04  sec.  (Falmouth  long.)  we  get 
G.  M.  T.,  7  hrs.,  10  min.,  12  sec.  At  the 
instant  of  obs.  chronom.  showed  7  hrs.,  14 
min.,  18  sec.  Chronom.  fast  of  G.  M.  T., 
4  min.,  06  sec.  May  28  obs.  showed  chronom. 
fast  of  G.  M.  T.  4  min.,  09.2  sec. 


146      ELEMENTS    OF   NAVIGATION 

May  20 4  m.  06  s. 

May  28 4  m.  09.2  s. 

Gain  in  8  days 3.2  a. 

Daily  rate  3  . 2 

=    .4  sec. 

8 

Other  celestial  bodies  can  be  used  as  well 
as  the  sun.  In  many  ports  chronometers 
may  be  rated  by  public  time  signals,  such  as 
time-balls  or  guns. 


CARE    OF   A    CHRONOMETER 

(Condensed  by  permission  of  T.  S.  and  J.  D.  Negus,  from  their 
paper  read  before  the  Naval  Institute.) 

Be  careful  in  carrying  a  chronometer  never 
to  give  it  a  horizontal  twist.  This  motion  will 
affect  the  balance  to  such  an  extent  as  to 
throw  the  chronometer  a  second  or  a  second 
and  a  half  out  of  time. 

The  gimbals  must  be  secured  so  as  to  pre- 
vent the  chronometer  from  swinging  while 
being  carried.  There  is  a  stay  for  this  pur- 
pose. Aboard  ship  the  instrument  should  be 
allowed  to  swing. 

Keep  a  chronometer  aboard  ship  always 
in  its  outside  case,  in  an  apartment  well 
ventilated,  yet  free  from  draughts.  Never 
put  a  chronometer  near  wood  which  is  in 
contact  with  salt-water. 

Never  open  the  outside  case  except  when 
winding  or  taking  time. 


CARE  OF  A  CHRONOMETER     147 

In  damp  countries  wrap  a  blanket  around 
the  outside  case. 

You  cannot  do  too  much  to  protect  a 
chronometer  from  rust.  A  small  spot  will 
change  the  rate  of  the  instrument. 

Wind  the  chronometer  every  day  at  the 
same  hour,  unless  it  is  an  eight-day  chro- 
nometer; then  wind  it  once  every  week  at 
the  same  time. 

In  winding,  turn  the  chronometer  bowl 
over  in  the  gimbal  slowly  with  the  left  hand, 
slide  the  valve  by  pressing  the  forefingers  of 
the  left  hand  against  the  nailpiece  on  the 
valve  until  the  key-hole  is  uncovered,  insert 
the  winding-key  with  the  right  hand,  and 
wind  to  the  left  till  a  decided  stop  is  felt. 
After  removing  the  key,  do  not  let  the 
chronometer  of  its  own  accord  drop  to  its 
level,  but  let  it  down  carefully  until  hori- 
zontal. 

Never  let  a  chronometer  get  within  the 
magnetic  influence  of  a  compass  or  an  electro- 
magnet. 

If  a  chronometer  has  run  down  and  needs 
to  be  started,  wait  till  the  hands  indicate 
the  proper  time,  and  then  start  it  by  a  slight 
horizontal  twist. 

All  chronometers  reach  their  highest  gain- 
ing or  losing  rate  at  a  certain  temperature. 
Those  used  in  the  United  States  Navy,  made 
by  Negus,  reach  their  fastest  rate  at  70°  F. 
Any   exposure   of   the   instrument   to   other 


148      ELEMENTS   OF   NAVIGATION 

temperatures  will  change  the  rate.  The 
average  temperature  correction,  as  given  by 
the  makers,  is  .0025  second,  multiphed  by 
the  square  of  the  difference  in  the  number  of 
degrees  of  temperature.  Thus,  to  find  the 
correction  to  be  made  to  the  rate  of  a  chro- 
nometer in  a  temperature  of  80°,  multiply 
.0025  by  the  square  of  the  difference  between 
70°  and  80°.  A  chronometer  with  a  rate  of 
+  1  sec.  at  70°  would  show  the  following 
variations : 


55°  60°  65°        70° 

+  .4375  s.      +  .75  3.     +  .9375  s.     +  1  a. 

75°        80°        85° 
+  .9375  s.    +  .75  s.    +  .4375  s. 


Chronometers  should  be  cleaned  and  oiled 
at  least  once  every  three  years  and  a  half. 

Vessels  destined  for  long  voyages  should 
carry  three  chronometers.  If  you  have  two 
and  one  goes  wrong,  you  cannot  tell  which 
is  in  error.  With  three  you  can  make  daily 
comparisons. 

Keep  your  chronometers  awaj'  from  iron. 
It  affects  their  rate. 

In  carrying  a  chronometer,  use  the  leather 
strap  on  the  case.  Do  not  swing  the  instru- 
ment or  let  it  be  knocked.  To  transport 
overland  (by  rail,  for  instance)  put  the  chro- 
nometer in  a  basket  on  plenty  of  cotton  or 
something  else  that  will  prevent  jarring. 


THE  DAY'S  WORK  149 


THE   DAY  S   WORK 

Before  leaving  port  ascertain  the  exact 
draught  of  your  vessel.  Also  ascertain  the 
height  of  your  eye  above  the  water-line  at 
all  points  available  for  taking  observations. 

As  soon  as  you  are  on  open  water  fix  the 
position  of  the  ship  by  cross-bearings,  by 
vertical  or  horizontal  angle  and  compass 
bearing,  or  by  compass  and  range-finder. 

This  is  called  taking  departure,  and  is 
entered  in  the  log  opposite  the  hour  thus: 
"Sandy  Hook  Lightship  bearing  S.  15°  W., 
distant  2  miles,  from  which  I  take  departure." 

From  the  moment  of  taking  departure 
begin  the  record  of  the  course  and  distance 
for  each  hour  in  the  log-l3ook.  Note  reading 
of  pat.  log  whenever  course  is  changed  or  a 
sight  taken. 

In  all  your  work  make  it  an  invariable 
practice  to  write  the  name  or  initials  of 
each  item  in  a  formula,  as  T.  C.  A.  (or  h, 
the  sj^mbol),  secant,  cos.,  etc.  This  will  save 
you  frequent  confusion  and  often  error. 

Ascertain  at  night  from  azimuth  tables  the 
hour  when  the  sun  will  bear  most  nearly 
east  next  morning. 

For  this  purpose  local  app.  time  need  be 
known  only  approximately. 

To  ascertain  watch  time  for  taking  morning 
sight  compare  watch  with  chronometer  the 


150      ELEMENTS    OF   NAVIGATION 

■  night  before.  Get  from  N.  A.  sun's  dec.  for 
next  A.M.  and  work  up  lat.  by  D.  R.  With 
these  enter  azimuth  tables  and  find  right  time 
to  take  morning  sight.  An  example  will  show 
the  rest  of  the  work. 

July  18,  1918,  you  find  that  next  morning 
about  8  A.M.  your  lat.  will  be  35°  10'  N., 
long.  60°  12'  W. 

Azimuth  table,  lat.  35°,  dec.  21°,  shows  that 
sun  win  bear  89°  34'  at  8  hrs.,  10  min.  a.m. 


Cor.  chronom.  reading  at  time  of  watch 

comparison 12  h.  00  m.  20  s. 

Equation  of  time —  6  m.  01  s. 


G.  A.  T.  at  comparison 11  h.   54  m.   19  s. 

Long.  60°  12'  W.   =  4  h.  00  m.  48  s 4  h.  00  m.  48  s. 


L.  A.  T.  at  time  of  comparison 7  h.   53  m.  31  s. 

Watch  T.  at  time  of  compari.son 7  h.  34  m.  22  s. 

Watch  wrong  on  L.  A.  T.  for  8  a.m.  sight  19  m.  09  s.  slow 

Time  of  89.°  34'  azimuth 8  h.   10  m.  00  s. 


Watch-time  to  take  a.m.  sight 7  h.  50  m.  51  s. 

Hence  take  the  sight  when  the  watch  shows 

7  hrs.,  51  min.,  and  the  sun  will  be  on  the 
P.  V.  When  the  watch  shows  7  hrs.,  51  min., 
the  correct  L.  A.  T.  will  be  8  hrs.,  10  min., 

08  sec. 

In  morning  note  comparison  of  hack  watch 
or  hack  chronometer  with  standard  chro- 
nometers. 

Ascertain  index  error  of  sextant. 

Take  a.m.  long,  sights  at  time  determined 
as  above.    Always  take  at  least  three  sights 


THE   DAY'S   WORK  151 

at  equal  intervals  of  time.  Use  mean  of 
altitudes  and  times  in  working  out.  This 
reduces  possible  errors. 

Although  the  ship  can  be  navigated  by 
St.  Hilaire  method,  at  least  one  chronometer 
sight  should  be  taken  each  day  to  keep  the 
local  time  accurately  checked  up.  This  is 
important. 

At  time  of  such  sight  take  bearing  of  sun 
by  standard  compass  and  ascertain  the  devi- 
ation. 

Set  ship's  clocks  by  L.  M.  T.  obtained  by 
applying  equation  to  L.  A.  'V.  computed 
from  sight. 

Lay  down  line  of  position  at  right  angles 
to  sun's  true  bearing.  If  any  line  of  position 
has  been  obtained  in  early  morning  hours 
run  it  up  and  get  ship's  position  as  already 
explained. 

If  running  at  high  speed,  scouting,  or  ap- 
proaching land,  get  other  lines  of  position  by 
St.  Hilaire  method  between  morning  and 
noon  sights. 

Ascertain  exact  run  of  ship  between  a.m. 
sight  and  noon,  and  set  clock  for  local  time 
of  noon  according  to  long.  To  do  this  re- 
member that  watch  must  be  set  back  for 
westerly  change  of  long,  and  forward  for 
easterly. 

Enter  Table  2  with  the  course  and  the 
hourly  speed  of  ship  as  dist.  Find  the  diff. 
long,   made   from   8   a.m.   sight  to   11    a.m. 


152      ELEMENTS   OF   NAVIGATION 

Apply  this  to  watch-time  to  ascertain  error 
of  watch  (W.)  at  11  a.m. 

For  example,  at  a.m.  sight  W.  was  19 
min.,  08  sec.  slow.  Suppose  your  change  of 
long-  to  11  A.M.  is  2  min.,  45  sec.  east,  then 
at  11  W.  is  21  min.,  53  sec.  slow,  and  the 
time  to  noon  will  be  1  hr.,  —  21  min.,  53  sec. 
=  38  min.,  07  sec.  Now  get  the  diif.  long, 
for  the  ship's  run  in  38  min.,  07  sec,  which 
is  the  change  between  11  a.m.  and  noon.  Let 
us  suppose  it  amounts  to  30  sec.  You  will 
have  this  result: 


Watch  slow  at  S  a.m.  sight 19  m.  OS  s. 

Change  to  1 1  a. m 2  m.  45  s. 

Change  from  11  to  L.  A.  noon 30  s. 


Total  change 22  m.  23  s. 

Run  to  noon,  4  hrs.   -  22  m.  23  s.   =    3  h.  37  m.  37  s. 

Run  from  a.m.  sight  to  noon    =   knots  per  hr. 

multiplied  by 3  h.  37  m.  37  s. 


Prepare  constant  for  noon  obs.  At  12  m. 
(app.  T.)  take  merid.  alt.  Run  up  latest 
Sumner  line  to  intersect  lat.  parallel  and  es- 
tablish noon  fix.  If  weather  is  cloudy,  take 
ex-merid.  before  noon  in  case  sun  is  covered 
at  noon. 

If,  following  the  old  method  of  taking  only 
the  A.M.  and  noon  sights  and  bringing  up  the 
long,  to  noon  by  D.  R.,  you  may  find  your 
D.  R.  lat.  considerablv  out.  In  this  case 
enter  Table  47  with  the  D.  R.  lat.  at  the 
top  and  the  azimuth  of  obs.  body  at  side, 
and  take  out  correction  called  longitude  fac- 


THE  DAY'S  WORK  153 

tor.  Multiply  it  by  diff.  between  lat.  by 
D.  R.  and  lat.  by  obs.  Result  is  correction 
to  be  applied  to  a.m.  long.,  which  can  then 
be  carried  forward  to  noon  by  D.  R.  If 
diff.  between  lats.  is  of  same  name  (N.  or  S.) 
as  first  letter  of  azimuth,  alteration  of  long, 
must  be  made  in  the  direction  contrary  to 
that  of  second  letter  of  azimuth  (E.  or  W.). 

Fix  by  intersecting  Sumner  lines,  located 
by  St.  Hilaire  method,  does  not  require  this 
correction. 

Owing  to  inevitable  errors,  a  vessel's  posi- 
tion is  rarely  determined  within  two  miles. 
Therefore  draw  a  circle  with  a  radius  of  2 
miles,  and  regard  the  ship  as  possibly  any- 
where within  it.  Plot  next  course  from  cir- 
cumference. 

After  obtaining  correct  noon  position,  com- 
pute course  and  distance  made  good  in  day's 
run. 

Compute  total  course  and  dist.  from  port 
of  departure;  also  from  port  of  destination. 
Difference  between  position  by  D.  R.  and 
that  by  obs.  alt.  is  usually  attributed  to 
current;  errors  in  steering,  etc.,  however,  are 
as  much  responsible. 

In  the  afternoon  work  time  sight  when  the 
sun  bears  most  nearly  west.  Or  lay  down 
afternoon  position  lines  by  St.  Hilaire  method. 

As  soon  as  stars  are  visible  try  to  get  a 
good  fix  by  simultaneous  observations  of 
two,   or   star    and   planet.     When   possible 


154      ELEMENTS   OF   NAVIGATION 

continue  this  work  at  intervals  through  night 
watches. 

Charts  expressly  made  for  plotting  posi- 
tions can  be  obtained.  They  save  the  sailing- 
chart  from  pencil-marks  and  rubber-smudges. 

Before  approaching  land  acquaint  your- 
self with  lights,  fog  signals,  soundings,  buoys, 
etc.,  as  shown  on  chart. 

Be  ready  to  recognize  any  light  as  soon 
as  seen.  If  flashing,  time  the  length  of  flash 
and  length  of  interval  when  light  is  still 
distant.  This  will  aid  in  identification,  and 
sometimes  make  it  certain. 

Before  entering  a  harbor  note  ranges, 
length  of  courses  to  be  steered  between 
turning-points,  etc.  If  danger  angles  have 
to  be  used,  plot  them  beforehand. 

Change  course  at  precise  turning-point. 
Note  time  and  read  patent  log. 

If  weather  is  thick,  steer  from  buoy  to 
buoy  along  channel,  allowing  for  tidal  cur- 
rent. If  you  fail  to  make  a  buoy  at  the  com- 
puted time,  anchor  at  once. 

On  reaching  your  anchorage,  plot  your 
position  on  chart  by  two  or  three  charted 
objects  whose  bearings  give  well-defined  in- 
tersections. 

COMPENSATION   OF  THE   COMPASS 

Under  certain  conditions  the  magnetic 
force  of  the  earth  communicates  magnetism 


COMPENSATION   OF   COMPASS     155 

to  iron  by  what  is  called  induction.  Ham- 
mering the  metal  causes  some  of  this  mag- 
netism to  persist,  and  what  remains  is  called 
sub-permanent  magnetism.  This  kind  of 
magnetism  originates  in  a  ship  while  she  is 
building,  and  is  most  potent  in  the  line  of  the 
earth's  magnetic  poles.  A  ship  built  with 
her  keel  magnetic  north  and  south  would 
have  her  magnetic  north  pole  at  the  bow 
and  south  at  the  stem. 

The  earth  exerts  also  a  horizontal  and  a 
vertical  magnetism.  The  former  is  most 
powerful  at  the  magnetic  equator  and  least 
so  at  the  magnetic  pole,  while  the  latter  is 
strongest  at  the  pole  and  zero  at  the  equa- 
tor. Magnetism  induced  by  these  forces  af- 
fects horizontal  and  vertical  soft  iron,  re- 
spectively, and  is  transient. 

The  results  of  the  operation  of  these  forces 
are  deviations  of  the  compass.  These  are  of 
three  kinds: 

Semicircular  dev.  is  due  to  the  sub-per- 
manent magnetism  of  the  ship  and  induced 
magnetism  of  soft  iron.  At  some  point  on 
the  compass -card  this  force  becomes  nil. 
If  you  swing  your  ship  to  the  right  from  this 
point  you  will  find  easterly  dev.  through  the 
first  semicircle  and  westerly  through  the 
second,  till  the  zero  point  is  reached  again. 
Hence  semicircular  dev.  is  that  in  which  the 
cause  operates  equally  in  opposite  directions 
through  the  two  semicircles. 


156      ELEMENTS   OF   NAVIGATION 

Quadrant  al  dev.  is  that  induced  in  soft 
iron  by  the  earth's  horizontal  magnetic  force. 
Deviations  of  this  class  change  their  sign 
every  90°  and  are  hence  named  quadrantal. 

Constant  dev.  is  due  to  nduction  in  hori- 
zontal soft  iron  irregularly  placed  in  relation 
to  the  compass.    There  is  also  a  heeling  error. 

Semicircular  dev.  is  corrected  by  magnets 
placed  athwartships  and  fore  and  aft  near 
the  compass.  The  binnacle  has  compart- 
ments for  these  magnets.  They  have  one 
end  red  and  the  other  blue,  and  you  must 
remember  that  the  latter  attracts  the  north 
end  of  the  needle. 

Quadrantal  dev.  is  corrected  by  placing 
hollow  spheres  of  soft  iron  at  equal  distances 
on  both  sides  of  the  compass.  Constant  dev. 
is  not  compensated  because  it  is  usually 
immaterial. 

See  that  the  ship  is  on  an  even  keel, 
compass  accurately  centered  in  binnacle,  all 
masses  of  iron  or  steel  near  compass  in  their 
customary  positions,  and  all  compensating 
correctors  taken  away. 

Place  the  ship's  head  on  N.  or  S.  correct 
magnetic.  For  this  purpose  use  the  Pelorus. 
Set  its  lubber's  point  at  N.  Clamp  the  sight- 
vanes  at  the  magnetic  bearing  (true  bearing 
corrected  to  include  variation)  of  the  observed 
object,  as  shown  by  chart.  Starboard  or 
port  helm  till  the  observed  object  (whether 
distant   one   on   shore   or  the   sun)    is  seen 


COMPENSATION  OF  COMPASS     157 

through  the  vanes.  The  ship  is  now  head- 
ing N.,  magnetic.  The  difference  between 
the  ship's  head  and  N.  on  the  compass-card 
is  the  deviation,  E.  if  comp.  bearing  is  to 
right  of  true,  W.  if  it  is  to  left.  Thus,  when 
Pelorus  shows  head  to  be  N.,  if  comp.  shows 
head  to  be  N.  by  E.,  there  is  one  point  E.  dev. 
In  using  the  sun  the  bearings  may  be  cal- 
culated beforehand  by  use  of  azimuth  tables 
and  L.  A.  T. 

Correct  with  thwartship  magnets.  Note 
whether  N.  point  is  drawn  toward  starboard 
or  toward  port.  If  to  starboard,  place  mag- 
net with  blue  end  to  port,  and  nice  versa. 

Having  corrected  N.  or  S.,  place  ship's 
head  E.  or  W.,  again  by  use  of  Pelorus.  Cor- 
rect with  fore  and  aft  magnets.  If  N.  is 
drawn  toward  bow,  blue  ends  of  magnets  go 
toward  stern,  and  vice  versa. 

Place  ship's  head  on  the  two  remaining 
cardinal  points  and  correct  half  the  error 
found  there  by  readjusting  the  magnets.  Do 
not  try  to  correct  all  or  you  will  throw  it 
back  on  the  first  two.  If  at  any  time  when 
you  have  placed,  say,  three  magnets  point- 
ing one  way,  they  pull  a  little  too  much,  try 
turning  the  lowest  one  the  opposite  way. 

This  completes  the  semi-circular  correc- 
tions. To  correct  quadrantal  error,  put  ship's 
head  N.E.,  S.E.,  S.W.,  N.W.,  and  get  devi- 
ation on  each.  Mark  E.  deviations  +  and 
W.  -. 


158      ELEMENTS   OF   NAVIGATION 

Reverse  the  sign  of  deviations  found  on  S.E. 
and  N.W.  Then  add  deviations  having  same 
sign,  take  difference  between  plus  and  minus 
quantities  and  prefix  sign  of  greater  to  result. 
Divide  this  result  by  4,  retaining  sign.  You 
now  have  the  quadrantal  dev.,  which  is  al- 
most always  +.  Unless  the  construction  of 
the  ship  be  changed  or  she  loads  with  iron, 
this  dev.  will  not  change. 

Example:  Dev.  on  N.  E.  -6°,  on  S.E.  - 
62°,  on  S.W.  +  32°,  on  N.W.  +  48° 


N.E. 

N.W. 

-  c,° 

-  48°  (sign  reversed) 

-  64° 

S.E.    +  62°  (sign  reversed) 
S.W.  +  32° 

+  94° 
-  54° 

4)  +  40° 

+  10°  =  Quad.  dev. 

To  compensate,  when  sign  is  +  put  ship's 
head  as  many  degrees  to  the  left  as  the  quad- 
rantal dev.  shows.  Keep  her  steady  and 
move  the  spheres  in  or  out  till  dev.  disap- 
pears. 

Heeling  error  is  corrected  by  a  vertical  bar 
in  a  tube  inside  the  binnacle.  The  scien- 
tific correction  requires  mathematical  cal- 
culations beyond  the  purpose  of  this  book. 
At  sea  when  the  ship  is  rolling  correct  error 
by  slowly  raising  or  lowering  bar  till  abnormal 
vibration  of  card  ends.  In  a  sailing-vessel, 
heeling  one  way  for  some  time  under  canvas, 
error  may  be  determined  by  an  azimuth. 


FINDING  THE  DEVIATION      159 

A  Flinders  bar  (a  bundle  of  soft  iron  rods 
in  a  case)  secured  vertically  near  the  com- 
pass, is  used  to  counteract  the  effect  of  change 
in  inductive  magnetism  of  soft  iron  in  chang- 
ing ship's  latitude.  It  can  be  placed  most 
accurately  when  the  ship  is  on  the  equator. 

Compasses  show  a  tendency  to  hang  back 
when  one  course,  especially  easterly  or  west- 
erly, is  steered  a  long  time.  If  you  steer  W. 
a  long  time,  expect  unusual  W.  error  if  you 
turn  to  N,,  or  E.  error  if  you  turn  S.  The 
same  principle  applies  to  E.  courses.  This 
sluggishness  of  the  compass  is  increased  by 
gun-fire.* 


FINDING    THE    DEVIATION 

In  order  to  compensate  a  compass  the  devi- 
ations existing  on  all  courses  must  be  ascer- 
tained. The  methods  by  which  they  are 
found  arc  used  again  to  determine  the  devi- 
ations remaining  after  compensation.  There 
are  four  methods,  each  of  which  requires  the 
ship  to  be  swung  slowly  around  a  circle,  by 
her  own  steam,  by  a  tug,  or  at  anchor  either 
by  the  tide  or  by  springs  and  hawsers.  All  ob- 
servations should  be  made  from  the  standard 

♦For  more  details,  consult  Bowditch,  articles  83-129;  Lecky's 
Wrinkles  in  Practical  Nnvitjnlinn,  chap,  xii;  The  A  B  C  of  Com- 
pass Adjustment,  by  E.  W.  Owens;  Instrurtions  for  the  Ailjust- 
inent  of  Lord  Kelvin's  Patent  Compass;  and  Compass  Adjustments, 
by  Lieut.  Win.  Appelbye-Robinson,  U.  S.  N.  R.  F. 

11 


160      ELEMEISTS   OF   NAVIGATION 

compass  and  the  others  may  be  corrected  at 
the  same  time  by  comparison  with  it. 

In  a  new  iron  or  steel  ship  observations 
must  be  made  every  15°  all  the  way  around. 
In  subsequent  corrections,  or  in  a  wooden 
ship,  observations  may  be  made  on  one-half 
or  one-fourth  the  number.  On  each  course 
the  ship  must  be  steadied  for  three  or  four 
minutes  so  that  the  card  may  come  to  rest 
and  magnetic  conditions  be  settled. 

1. — Reciprocal  bearings.  One  observer  on 
shore  sets  a  compass  where  there  is  no  local 
magnetic  influence;  the  other  observer  is  at 
the  ship's  standard  compass.  On  each  course, 
as  the  ship  swings,  the  observers  at  signal 
take  each  other's  bearing.  The  reverse  of 
the  shore  observer's  bearing  of  the  ship  ob- 
server is  the  magnetic  (having  variation, 
but  no  deviation)  bearing  of  the  man  on 
shore.  The  difference  between  this  and  the 
bearing  shown  by  the  standard  compass  is 
the  dev.  The  observers  should  compare 
watches  before  beginning  and  time  each  ob- 
servation to  check  records. 

2. — By  ranges.  Two  range  marks,  such 
as  the  Swash  Channel  lights,  New  York  Bay, 
whose  magnetic  bearing  is  charted,  can  be  used 
for  compass  comparison. 

4. — By  true  bearing  of  celestial  body. 
For  the  sun  the  L.  A.  T.  and  for  a  star, 
planet,  or  the  moon  the  H.  A.  must  be  used 
in  entering  the  azimuth  tables,   as  already 


FINDING   THE  DEVIATION      161 

explained.  If  the  celestial  body  has  a  decli- 
nation greater  than  those  contained  in  the 
tables  its  azimuth  may  be  computed  by  the 
azimuth  formula  or  ascertained  from  Table 
V  in  H.  O.  Book  No.  200,  for  the  use  of  which 
very  clear  directions  are  given. 

Alt. -azimuth  formula:  add  P.  D.,  lat.  and 
T.  C.  A.  and  find  half-sum  as  in  time  sight. 
Diff.,  however,  is  now  P.  D. — half-sum  (in- 
stead of  T.  C.  A.).  Add  secant  P.  D.,  secant 
lat.,  cosine  half-sum,  and  cosine  diff.  This 
gives  cosine  of  half  the  azimuth,  so  multiply 
by  2  to  get  correct  angle  of  bearing. 

If  the  time  is  accurately  known,  you  may 
use  this  formula:  Sine  H.  A.  +  cosine  dec. 
-1-  secant  alt.  =  sine  azimuth. 

By  a  distant  object. — Select  a  well-defined 
object  on  shore  at  least  one  hundred  times  as 
far  away  from  the  ship  as  the  diameter  of 
her  swing.  Obtain  its  magnetic  bearing  from 
the  chart,  or  by  taking  a  compass  ashore  and 
placing  it  in  range  between  the  center  of  the 
ship's  swing  and  the  object,  or  by  swinging 
ship  and  accepting  the  mean  of  all  the  bear- 
ings as  the  magnetic.  Then  compare  com- 
pass bearings  with  this  magnetic  bearing. 
Observations  should  be  tabulated  for  inspec- 
tion and  reference  in  some  such  manner  as 
the  following: 

Observations  for  series  of  deviations,  Aug. 
10,  1891,  at  Coquimbo,  Chile.  Object  ob- 
served,   mountain.      Magnetic    bearing    of 


162      ELEMENTS   OF   NAVIGATION 

mountain  from  mean  of  all  compass  bearings, 
N.  3°  04'  W.* 


Swinging  Port  to  Starboard 


Ship's  Head 

by 

Standard  Comp. 


Bearing  of 

Mountain  by 

Standard  Comp. 


Deviation 


N. 

N.  by  E. 

N.  N,  E. 
N.  E.  by  N. 
N.  E. 


N.  4°   10'  W. 

7°  15' 

10°  15' 

12°  20' 

13°  40' 


1°  30'  E. 
4°  35' 
7°  35' 
9°  40' 
11°  00' 


This  tabulation  is  carried  out  through  the 
entire  swinging,  the  ship  being  taken  around 
first  from  port  to  starboard  and  then  the  op- 
posite way. 

THE    NAPIER    CURVE 

The  most  convenient  way  of  recording 
deviations  and  correcting  courses  is  by  the 
use  of  the  Napier  curve.  This  is  constructed 
by  representing  the  circumference  of  the  com- 
pass as  a  straight  line  and  marking  off  on  it  the 
degrees  from  0°  to  360°. 

Diagonally  downward  to  the  left  at  every 
15°  is  drawn  a  plain  line,  and  to  the  right  a 
dotted  line.  These  lines  are  inclined  to  the 
perpendicular  at  an  angle  of  60°. 

The  deviations  are  marked  off  with  a  curve 
as  in   diagram   on   page   162.    In   swinging 


*  From  Practical  Problems  and  the  Com,pensation  of  the  Compass 
in  the  U.  S.  Navy.     Pub.  by  Navy  Dept.,  1898. 


THE  NAPIER  CURVE  163 

ship  on  every  15°,  as  before  described,  the 
deviation  obtained  on  each  course  is  meas- 
ured off  on  the  vertical  scale  and  then  laid 
off  on  one  of  the  lines  passing  through  the 


FfomO'CN)  to  leo'Cs) 

Deviation  West 


This  diagram  shows  the  top  of  a  Napier  curve,  represented  by 
the  heavy  line,  and  the  diagonals  drawn  through  each  interval 
of  15°.  The  straight  perpendicular  represents  the  circumference 
of  the  compass-card.  I'roni  180°  to  31)0°  would  be  placed  to  the 
right  of  the  first  half.  Wc  show  here  only  the  first  quarter,  N.  to  E. 


164      ELEMENTS   OF   NAVIGATION 

point.  If  the  deviations  were  observed  on  a 
compass  course  mark  on  the  dotted  hne, 
right  if  E.,  left  if  W.  If  the  deviations  were 
observed  on  magnetic  courses,  mark  on  the 
plain  line. 

If  the  observations  were  not  made  on  a 
15°  division,  draw  a  parallel  line  through  the 
degree  representing  the  ship's  heading.  Mark 
each  point  thus  obtained  and  draw  a  curve 
through  them,  as  in  the  diagram,  Avhich  is 
used  as  follows: 

1.  Given  the  compass  course,  to  find  the 
corresponding  magnetic  course. — Place  one  leg 
of  a  pair  of  dividers  on  the  vertical  line  at  the 
given  compass  degree;  place  the  second  leg 
on  the  curve  where  it  intersects  the  dotted 
line  passing  through  the  given  point,  or  where 
it  intersects  a 'line  drawn  parallel  to  the  dotted 
line  through  the  given  course;  swing  the 
second  leg  from  the  curve  to  the  vertical 
line,  downward  if  the  deviation  be  easterly, 
upward  if  westerly;  the  point  where  the 
second  leg  touches  the  vertical  line  will  be 
the  magnetic  course  required. 

2.  Given  the  magnetic  course,  to  find  the 
corresponding  compass  course. — Place  one  leg 
of  a  pair  of  dividers  on  the  vertical  line  at 
the  given  magnetic  course ;  place  the  second 
leg  on  the  curve  where  it  intersects  the  plain 
line  passing  through  the  given  magnetic 
point,  or  where  it  intersects  a  line  drawn 
parallel  to  the  plain  line  through  the  given 


THE  NAPIER  CURVE  165 

point;  swing  the  second  leg  from  the  curve 
to  the  vertical  line,  upward  if  the  deviation 
be  easterly,  downward  if  westerly;  the  point 
where  the  second  leg  touches  the  vertical  line 
will  be  the  compass  course  required. 


166       ELEMENTS   OF   NAVIGATION 


EXAMPLES   FOR   PRACTICE 

DEAD-RECKONING 

Suppose  a  ship  to  sail  upon  the  following 
courses  and  distances:  S.E.-by-S.,  29  miles 
N.N.E.,  10;  E.S.E.,  50;  E.N.E.,  50;  S.S.E. 
10;  N.E.-by-N.,  29;  W.,  25;  S.S.E.,  10 
W.S.W.KW.,  42;  N.,  110;  E  3^N.,  62;  N. 
7;  W.,  62;  N.,  10;  W.,  8;  S,  10;  W.,  62 
S.,  7;  E.^S.,  62;  S.,  110;  W.N.W.i^W.,  42 
N.N.E.,  10;  and  W.,  25.  Required  the  course 
and  distance  made  good  (Norie). 

Ans.  The  ship  has  returned  to  the  place 
she  started  from. 

From  lat.  40°  3'  N.,  long.  73°  28'  W.,  ship 
sails  S.E.-by-S.,  36  miles,  variation  ^  pt. 
west;  S.E.-by-S.,  8  miles,  variation  34  pt. 
west;  S.E.J/^E.,  28  miles,  with  half  a  point 
of  leeway  on  the  starboard  tack  and  varia- 
tion 34  pt.  west.  Ship  has  been  8  hrs.  in  a 
current  setting  N.E.  (variation  34  pt-  W.) 
at  the  rate  of  2  knots  per  hr.  Required  lat. 
and  long,  in  and  course  and  distance  made 
good  (Patterson). 

Ans.  Lat.  39°  26'  N.,  long.  72°  07'  W., 
course  S.  60°  E.,  dist.  72  miles. 

At  9,15  A.M.  pat.  log  reading  15.3.      Lat. 


EXAMPLES  FOR  PRACTICE      167 

40°  28'  N.,  long.  73°  50'  W.  Course  per 
standard  compass,  159°.  Var.  11°  W.,  dev. 
3°  E.  until  noon,  when  pat.  log  reads  50.7. 
Noon  obs.  gives  lat.  40°  00'  N.,  long.  73° 
32'  W.  From  noon  to  4  p.m.  course  150°, 
var.  10°  W.,  dev.  4°  E.  Pat.  log  4  p.m., 
98.9.  Time  sight  4  p.m.  gives  long.  72°  59' 
W.  From  4  p.m.  to  8  p.m.  course,  var.  and 
dev.  the  same.  Pat.  log  8  p.m.,  145.4.  Re- 
quired D.  R.  position  8  p.m. 

Ans.  Lat.  38°  43'36  "  N.,  long.  72°  24'  W. 

Noon  position,  lat.  10°  15'  S.;  long.  150° 
47'  W.  Pat.  log  noon,  126.4.  Noon  to  4 
p.m.,  comp.  course  287°;  var.  14°  E.;  dev. 
3°  E.  Pat.  log  4  P.M.,  174.4.  Time  sight 
4  P.M.  gives  long.  151°  36'  W.  From  4  p.m. 
to  8  P.M.  comp.  course  300°;  var.  16°  E. ; 
dev.  5°  E.  Pat.  log  8  p.m.,  223.8.  Required 
8  P.M.  position  of  ship,  course,  and  dist.  made 
good  since  noon,  and  course  and  dist.  to  lat. 
2°  20'  S.,  long.  161°  27'  W. 

Ans.  Lat.  9°  10'  06"  N.,  long.  152°  07'  W. 
Course  and  dist.  since  noon,  309°,  104  miles. 
Course  and  dist.  to  go,  306°,  700  miles. 


SHAPING    COURSE   BY   MERCATOR  S   SAILING 

Required  the  bearing  and  distance  of  Per- 
nambuco,  lat.  8°  4'  S.,  long.  34°  53'  W.,  from 
Cape  Verde,  lat.  14°  45'  N.,  long.  17°  32'  W. 
(Norie). 


168       ELEMENTS   OF   NAVIGATION 

Ans.  S.  37°  W.  (217°),  dist.  1715  miles. 

Required  course  and  distance  from  Cape 
Palmas,  lat.  4°  24'  N.,  long.  7°  46'  W.,  to 
St.  Paul  de  Loando,  lat.  8°  48'  S.,  long.  13° 
8'  E.  (Norie). 

Ans.  S.  58°  E.  (122°),  dist.  1481  miles. 

LATITUDE    BY    MERIDIAN    ALTITUDE    OF    SUN 

At  sea,  merid.  alt.  O  38°  15'  15"  S.;  I.  E., 
1°  10'  -;  H.  of  E.,  15  ft.;  chronom.,  4  hrs., 
10  min.,  18  sec.  p.m.;  chronom.  slow  of  G. 
M.  T.,  4  min.,  37  sec;  dec,  4  p.m.,  15°  30' 
11"  N.,  increasing;  hourly  var.,  44.6".  Re- 
quired lat.  of  ship. 

Ans.  68°  14'  N. 

At  sea,  merid.  alt.  O  48°  18'  15"  N.;  I.  E., 
-  2'  15";  H.  of  E.,  20  ft.;  G.  M.  T.,  10  hrs., 
26  min.,  15  sec.  a.m.;  dec.  noon,  19°  26'  S.,  de- 
creasing; hourly  var.,  .6'.  Required  lat.  of 
ship. 

Ans.  61°  17'  S. 

LATITUDE   BY   MERIDIAN  ALTITUDE   OF  STAR 

At  sea,  Dec.  24,  1894.  Merid.  alt.  ^  Al- 
debaran  52°  36'  S.;  no  I.  E.;  H.  of  E.,  20 
ft.;  dec  of  ^  16°  17'  52"  N.  Required  lat. 
of  ship. 

Ans.  53°  4734'  N. 

At  sea,  Dec.  26,  1894.  Merid.  alt.  Sirius 
36°  28'   S.;  I.  E.,  -  45";  H.  of  E.,  14  ft.; 


EXAMPLES  FOR  PRACTICE      169 

doc.  of  *,  16°  34'  20"  S.      Required  lat.  of 
ship. 

Ans.  37°  3'  N. 

LATITUDE    BY    EX-MERIDIAN   ALTITUDES 

At  sea,  July  12,  1885.  Lat.  by  D.  R.  50° 
N.,  long,  by  D.  R.  40°  W.;  obs.  ex-merid. 
alt.  O  61°  48'  30";  L  E.,  -  3';  dip,  3'  48"; 
G.  M.  T.  of  obs.,  2  hrs.,  39  min.,  9  sec;  dec.  of 
O  2  P.M.,  21°  54'  18"  N.;  hourly  diff.  dec, 
.3', dec  decreasing;  equation  of  time  to  be  sub- 
tracted from  M.  T.,  5  min.,  20  sec;  hourly 
diff.  equation,  .3,  equation  decreasing.  Re- 
quired lat.  of  ship. 

Ans.  49°  56'  N. 


LATITUDE    BY   THE    POLE-STAR 

At  sea,  June  21,  1880.  Lat.  by  D.  R.  45° 
20'  N.,  long.  37°  57'  W.;  obs.  alt.  of  Polaris, 
44°  13'  30"  N.;  L  E..  +  30";  H.  of  E.,  32  ft.; 
G.  M.  T.,  11°  45'  20";  G.  Sid.  T.  preceding 
noon,  6  hrs.,  14  sec  Required  lat.  of  ship 
(Lecky). 

Ans.  45°  17'  N. 

LONGITUDE   BY   CHRONOMETER  SIGHT 

Observed  a.m.  alt.  O  20°  30';  chronom.  1 
hr.,  11  min.,  19  sec  p.m.;  chronom.  10  min., 


170       ELEMENTS   OF   NAVIGATION 

20  sec.  fast;  H.  of  E.,  10  ft.;  lat.  by  D.  R. 
40°  15'  N.;  dec.  at  noon,  13°  26'  6"  S.; 
hourly  diff.  dec,  .5',  dec.  decreasing;  equa- 
tion of  time,  14  min.,  28  sec;  hourly  diff. 
equation,  .05",  equation  decreasing;  equa- 
tion to  be  added  to  app.  time.  Required  long, 
of  ship  (Patterson). 

Ans.  58°  59'  45"  W. 

At  sea  Jan.  22,  1895.  Obs.  alt.  of  O  a.m. 
17°  14';  G.  M.  T.,  11  hrs.,  42  min.  a.m.; 
H.  of  E.,  20  ft.;  no  I.  E.;  lat.  38°  50'  N.; 
dec.  at  noon,  23°  33"  S.;  hourly  diff.,  12.48', 
dec  decreasing;  equation  of  time  (to  be  sub- 
tracted from  mean  time),  3  min.,  46.42  sec; 
hourly  diff.  equation,  1.2  sec,  equation  in- 
creasing.    Required  long,  of  ship. 

Ans.  Long.  34°  18'  30"  W. 

At  sea,  Feb.  27,  1882.  Lat.  40°  10'  45" 
N.;  H.  of  E.,  30  ft.;  no  I.  E.;  obs.  alt.  * 
Procyon,  39°  11'  E.;  G.  M.  T.,  9  hrs.,  58 
min.,  45  sec;  Sid.  T.  at  G.  at  preceding 
noon,  22  hrs.,  28  min.,  52  sec;  dec  ^,  5° 
31'  15"  N.;  R.  A.  ^,  7  hrs.,  33  min.,  10  sec. 
Required  long,  of  ship,  true  bearing  of  star, 
and  Sumner  line  (Lecky) . 

Ans.  Long.  55°  40'  15"  W.;  true  bearing 
of  star,  S.  58°  E.;  Sumner  line,  N.  32°  E. 


THE   END 


TMs  book  is  DU£  on  the  last  date  stamped  below 

SEP  3      1<l34       \f^^-^' 


APK6    iB38  gtei  6  2  iOO  LiscHARn,    ^. 


m"<^^ 


tStl^EC  2  11942^1 


)^ 


t^ 


1942 

Rf-'.-.   ..  -. 

m  £  ?  f979 


k^  d     IS4Bj 
SEP  2  3  1943/ 

JUNl  5  1950 
AUG  1  2  1954 


MAR  4     194^ 

orm  L-9-15m-7,'82 


2  1979 


-'i/b 


A     000  726  388     2^^ 


